Power-exponential velocity distributions in disordered porous media

Matyka, Maciej; Gołembiewski, Jarosław; Koza, Zbigniew

doi:10.1103/physreve.93.013110

Cited by 32 publications

(39 citation statements)

References 32 publications

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“…As the porosity increases, the distribution approaches a stretched exponential function for large v ,

p ((), \overset{v}{true}) \sim \exp ((), - α {\overset{v}{true}}^{β})

where

\overset{v}{true} = v / (〈〉, v)

, the exponent β ≃1/2, and the scale parameter α ≃0.25. This observation is consistent with established velocity statistics for disordered porous media [ Matyka et al , ], and the stretched exponential distribution can theoretically be inferred by considering the porous media as a collection of cylinders with exponentially distributed radii [ Holzner et al , ].…”

Section: Resultssupporting

confidence: 87%

“…With regard to the speed distributions presented in section 3, a stretched exponential probability density function provides a good fit for large speeds. A shifted, stretched exponential (power exponential) distribution, as proposed by Matyka et al [], would also be in agreement with our results, but this would require introducing another fitting parameter. Moreover, the evolving pore structure due to dissolution in our sample does not significantly alter the functional dependence of the probability density function of fluid velocities, when rescaled by the average velocity.…”

Section: Discussionsupporting

confidence: 91%

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Self‐similar distributions of fluid velocity and stress heterogeneity in a dissolving porous limestone

2017

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In a porous rock, the spatial distribution of the pore space induces a strong heterogeneity in fluid flow rates and in the stress distribution in the rock mass. If the rock microstructure evolves through time, for example, by dissolution, fluid flow and stress will evolve accordingly. Here we consider a core sample of porous limestone that has undergone several steps of dissolution. Based on 3‐D X‐ray tomography scans, we calculate numerically the coupled system of fluid flow in the pore space and stress in the solid. We determine how the flow field affects the stress distribution both at the pore wall surface and in the bulk of the solid matrix. We show that during dissolution, the heterogeneous stress evolves in a self‐similar manner as the porosity is increased. Conversely, the fluid velocity shows a stretched exponential distribution. The scalings of these common master distributions offer a unified description of the porosity evolution, pore flow, and the heterogeneity in stress for a rock with evolving microstructure. Moreover, the probability density functions of stress invariants (mechanical pressure or von Mises stress) display heavy tails toward large stresses. If these results can be extended to other kinds of rocks, they provide an additional explanation of the sensitivity to failure of porous rocks under slight changes of stress.

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“…As the porosity increases, the distribution approaches a stretched exponential function for large v ,

p ((), \overset{v}{true}) \sim \exp ((), - α {\overset{v}{true}}^{β})

where

\overset{v}{true} = v / (〈〉, v)

Section: Resultssupporting

confidence: 87%

Section: Discussionsupporting

confidence: 91%

Self‐similar distributions of fluid velocity and stress heterogeneity in a dissolving porous limestone

2017

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“…This phenomenon is responsible for incomplete mixing or enhanced spreading, persistently spanning scales from the pore to the field (Berkowitz et al, ; Dentz et al, ; Edery et al, ; Gouze et al, ; Le Borgne et al, ; Le Borgne & Gouze, ). At the pore scale, anomalous transport exhibits many different characteristics such as non‐Gaussian velocity distributions (Bijeljic et al, ; Matyka et al, ), high temporal correlation of Lagrangian velocities forming a spatial Markov process (Le Borgne et al, ), intermittency of velocities along trajectories (de Anna et al, ), and superdiffusive spreading (Holzner et al, ; Kang et al, ). The intensity of the anomalous transport is related to the heterogeneity of the porous medium and has been investigated in media of different complexities, ranging from simple beadpacks to fractured sandstones and carbonates (Bijeljic et al, ; Meyer & Bijeljic, ; Morales et al, ; Siena et al, ).…”

Section: Introductionmentioning

confidence: 99%

Pore‐Scale Hydrodynamics in a Progressively Bioclogged Three‐Dimensional Porous Medium: 3‐D Particle Tracking Experiments and Stochastic Transport Modeling

Carrel

Morales

Dentz

et al. 2018

Water Resources Research

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Biofilms are ubiquitous bacterial communities that grow in various porous media including soils, trickling, and sand filters. In these environments, they play a central role in services ranging from degradation of pollutants to water purification. Biofilms dynamically change the pore structure of the medium through selective clogging of pores, a process known as bioclogging. This affects how solutes are transported and spread through the porous matrix, but the temporal changes to transport behavior during bioclogging are not well understood. To address this uncertainty, we experimentally study the hydrodynamic changes of a transparent 3‐D porous medium as it experiences progressive bioclogging. Statistical analyses of the system's hydrodynamics at four time points of bioclogging (0, 24, 36, and 48 h in the exponential growth phase) reveal exponential increases in both average and variance of the flow velocity, as well as its correlation length. Measurements for spreading, as mean‐squared displacements, are found to be non‐Fickian and more intensely superdiffusive with progressive bioclogging, indicating the formation of preferential flow pathways and stagnation zones. A gamma distribution describes well the Lagrangian velocity distributions and provides parameters that quantify changes to the flow, which evolves from a parallel pore arrangement under unclogged conditions, toward a more serial arrangement with increasing clogging. Exponentially evolving hydrodynamic metrics agree with an exponential bacterial growth phase and are used to parameterize a correlated continuous time random walk model with a stochastic velocity relaxation. The model accurately reproduces transport observations and can be used to resolve transport behavior at intermediate time points within the exponential growth phase considered.

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“…Recent experimental and numerical studies have shown that the occurrence of non‐Fickian particle dispersion due to long advective residence times is directly linked to intermittency in the Lagrangian velocity time series (Carrel et al, ; De Anna et al, ; Holzner et al, ; Kang et al, ; Morales et al, ). Thus, the understanding of these phenomena requires a sound characterization and understanding of the dynamics of Lagrangian and Eulerian pore‐scale velocities, which have been the subject of a series of recent studies (De Anna et al, ; Gjetvaj et al, ; Holzner et al, ; Jin et al, ; Meyer & Bijeljic, ; Morales et al, ; Matyka et al, ; Siena et al, ).…”

Section: Introductionmentioning

confidence: 99%

Stochastic Dynamics of Lagrangian Pore‐Scale Velocities in Three‐Dimensional Porous Media

Puyguiraud

Gouze

Dentz

2019

Water Resources Research

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Upscaling dispersion, mixing, and reaction processes from the pore to the Darcy scale is directly related to the understanding of the dynamics of pore-scale particle velocities, which are at the origin of hydrodynamic dispersion and non-Fickian transport behaviors. With the aim of deriving a framework for the systematic upscaling of these processes from the pore to the Darcy scale, we present a detailed analysis of the evolution of Lagrangian and Eulerian statistics and their dependence on the injection condition. The study is based on velocity data obtained from computational fluid dynamics simulations of Stokes flow and advective particle tracking in the three-dimensional pore structure obtained from high-resolution X-ray microtomography of a Berea sandstone sample. While isochronously sampled velocity series show intermittent behavior, equidistant series vary in a regular random pattern. Both statistics evolve toward stationary states, which are related to the Eulerian velocity statistics. The equidistantly sampled Lagrangian velocity distribution converges on only a few pore lengths. These findings indicate that the equidistant velocity series can be represented by an ergodic Markov process. A stochastic Markov model for the equidistant velocity magnitude captures the evolution of the Lagrangian velocity statistics. The model is parameterized by the Eulerian velocity distribution and a relaxation length scale, which can be related to hydraulic properties and the medium geometry. These findings lay the basis for a predictive stochastic approach to upscale solute dispersion in complex porous media from the pore to the Darcy scale.

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Power-exponential velocity distributions in disordered porous media

Cited by 32 publications

References 32 publications

Self‐similar distributions of fluid velocity and stress heterogeneity in a dissolving porous limestone

Self‐similar distributions of fluid velocity and stress heterogeneity in a dissolving porous limestone

Pore‐Scale Hydrodynamics in a Progressively Bioclogged Three‐Dimensional Porous Medium: 3‐D Particle Tracking Experiments and Stochastic Transport Modeling

Stochastic Dynamics of Lagrangian Pore‐Scale Velocities in Three‐Dimensional Porous Media

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