Anomalous Transport in Random Fracture Networks

Berkowitz, Brian; Scher, H.

doi:10.1103/physrevlett.79.4038

Cited by 292 publications

(246 citation statements)

References 10 publications

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“…In addition to using the commonly applied AD and single-rate MIM models, the SC and CTRW models will be applied. CTRW is tested since this model is reported to have the potential to describe a range of transport behaviors, and the coupled CTRW proposed by Berkowitz and Scher [1997] and has not previously been applied to unsaturated transport.…”

Section: W07402mentioning

confidence: 99%

Non‐Fickian transport in homogeneous unsaturated repacked sand

Bromly

Hinz

2004

Water Resources Research

136

108

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[1] It is commonly assumed a priori that solute transport experiments conducted in homogeneously repacked laboratory columns can be described by the advective-dispersive (AD) or single-rate mobile-immobile (MIM) transport model. To investigate this, nonreactive transport through macroscopically homogeneous repacked unsaturated sand was studied at two water contents using laboratory columns of diameter of 11 cm and lengths of 10, 20, and 40 cm. Non-Fickian behavior was found to dominate transport at this scale, with long breakthrough curve (BTC) tailing persistent over the range of column lengths tested. Measured BTC tails were well described by the single-rate MIM model at each depth; however, over the range of travel distances studied BTC tailing was better explained as a result of stochastic-convective (SC) transport or continuous time random walk (CTRW). The SC model was applied using the BTC measured at 10 cm as the probability density function to make predictions to subsequent depths. The CTRW model spreading parameter (b) remained approximately constant across the range of both travel distances and water contents considered. It is concluded that the assumption of a single-rate MIM model cannot be made a priori for macroscopically homogeneous unsaturated sands. In this case, variability between replicates prevented identification of whether a SC transfer function or CTRW provided the best description of transport. These results demonstrate the variability in transport as a result of heterogeneities in column packing, even for macroscopically homogeneous sand, and emphasize the importance of studying transport over a range of travel distances in order to allow prediction of transport.

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Section: W07402mentioning

confidence: 99%

Non‐Fickian transport in homogeneous unsaturated repacked sand

Bromly

Hinz

2004

Water Resources Research

136

108

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“…Meerschaert et al, 1999] have as fundamental solutions the stable distributions, and these results provide independent support for this approach from a mechanistic process level. With appropriate choice of parameters, the continuous time random walk [Berkowitz and Scher, 1997] and Boltzmann equation [Williams, 1992] approaches should also be able to represent the relevant phenomenology in the non-reacting situation.…”

Section: Implications and Conclusionmentioning

confidence: 99%

Power‐law velocity distributions in fracture networks: Numerical evidence and implications for tracer transport

Painter

Cvetković

Selroos

2002

Geophysical Research Letters

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[1] Velocity distributions in two-and three-dimensional networks of discrete fractures are studied through numerical simulations. The distribution of 1/v, where v is the velocity along particle trajectories, is closely approximated by a power law (Pareto) distribution over a wide range of velocities. For the conditions studied, the power law exponents are in the range 1.1-1.8, and generally increase with increasing fracture density. The same is true for the quantity 1/bv, which is related to retention properties of the rock; b is the fracture half-aperture. Using a stochastic Lagrangian methodology and statistical limit theorems applicable to power-law variables, it is shown that the distributions of residence times for conservative and reacting tracers are related to one-sided stable distributions. These results are incompatible with the classical advection dispersion equation and underscore the need for alternative modeling approaches.

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“…It was verified that its corresponding equilibrium solution is given by a q-exponential for an arbitrary confining potential. The theoretical background for most physical applications related to nonlinear Fokker-Planck equations, like the motion of particles in porous media [12][13][14][15], the dynamics of surface growth [15], and the dynamics of interacting vortices in disordered superconductors [17,18], which usually take place in physical spaces of dimensions 2 and 3, was presented.…”

Section: Discussionmentioning

confidence: 99%

“…These type of phenomena may be found in the motion of particles in porous media [12][13][14][15], the dynamics of surface growth [15], the diffusion of polymer-like breakable micelles [16], the dynamics of interacting vortices in disordered superconductors [17,18], and the motion of overdamped particles through narrow channels [19], among others. An interesting aspect about the Tsallis distribution is that it appears also as a a stable solution of a NLFPE; in its simplest, one-dimensional form, this equation is given by [20][21][22] …”

Section: Introductionmentioning

confidence: 99%

Classes of N-Dimensional Nonlinear Fokker-Planck Equations Associated to Tsallis Entropy

Ribeiro

Nobre

Curado

2011

Entropy

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Several previous results valid for one-dimensional nonlinear Fokker-Planck equations are generalized to N-dimensions. A general nonlinear N-dimensional FokkerPlanck equation is derived directly from a master equation, by considering nonlinearities in the transition rates. Using nonlinear Fokker-Planck equations, the H-theorem is proved; for that, an important relation involving these equations and general entropic forms is introduced. It is shown that due to this relation, classes of nonlinear N-dimensional Fokker-Planck equations are connected to a single entropic form. A particular emphasis is given to the class of equations associated to Tsallis entropy, in both cases of the standard, and generalized definitions for the internal energy.

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Anomalous Transport in Random Fracture Networks

Cited by 292 publications

References 10 publications

Non‐Fickian transport in homogeneous unsaturated repacked sand

Non‐Fickian transport in homogeneous unsaturated repacked sand

Power‐law velocity distributions in fracture networks: Numerical evidence and implications for tracer transport

Classes of N-Dimensional Nonlinear Fokker-Planck Equations Associated to Tsallis Entropy

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