Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications

Zhang, Yong; Benson, David A.; Reeves, Donald M.

doi:10.1016/j.advwatres.2009.01.008

Cited by 305 publications

(238 citation statements)

References 143 publications

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“…A random walk with a L´ vy distribution of jump lengths is described on large scales by a diffusion equation with a Riesz fractional derivative. Some space Riesz fractional diffusion equations have been proposed and used in natural systems including heterogeneous soils, aquifers, and rivers, is typically observed to be non-Fickian, also called anomalous (see [23]). A onecomponent system is governed by the equation…”

Section: Introductionmentioning

confidence: 99%

Numerical simulation for two-dimensional Riesz space fractional diffusion equations with a nonlinear reaction term

Liu¹,

Chen²,

Turner³

et al. 2013

Open Physics

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Abstract:Fractional differential equations have attracted considerable interest because of their ability to model anomalous transport phenomena. Space fractional diffusion equations with a nonlinear reaction term have been presented and used to model many problems of practical interest. In this paper, a two-dimensional Riesz space fractional diffusion equation with a nonlinear reaction term (2D-RSFDE-NRT) is considered. A novel alternating direction implicit method for the 2D-RSFDE-NRT with homogeneous Dirichlet boundary conditions is proposed. The stability and convergence of the alternating direction implicit method are discussed. These numerical techniques are used for simulating a two-dimensional Riesz space fractional Fitzhugh-Nagumo model. Finally, a numerical example of a two-dimensional Riesz space fractional diffusion equation with an exact solution is given. The numerical results demonstrate the effectiveness of the methods. These methods and techniques can be extended in a straightforward method to three spatial dimensions, which will be the topic of our future research.PACS (2008)

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Section: Introductionmentioning

confidence: 99%

Numerical simulation for two-dimensional Riesz space fractional diffusion equations with a nonlinear reaction term

Liu¹,

Chen²,

Turner³

et al. 2013

Open Physics

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“…Note that these two processes are independent, driven by different hydraulic properties. The mixed super-and sub-diffusion is not uncommon for transport through real media [12]. The simulation result generally captures the two tails of the observed product snapshot (Fig.…”

Section: Applications: Lagrangian Modeling Of the Bimolecular Reactivmentioning

confidence: 64%

“…To illustrate the physical process, we build the continuum model. The space fractional-order PDE [7,17,18] can efficiently capture super-diffusive transport of nonreactive tracers observed in complex, natural geological formations, as reviewed and demonstrated further by Zhang et al [12]. The space tempered stable model [19,20,21] is the logic extension of the standard fractional-order PDE to capture the convergent spatial moments and the natural cutoff of power-law distributions present in real physical systems.…”

Section: Lagrangian Simulation Of Bimolecular Reaction Controlled By mentioning

confidence: 99%

“…However, model (17) describes the trapping of solute particles in immobile domains, representing a delayed transport. Hence, for description simplicity, we name the resultant transport as a subdiffusive process, to keep consistent to the terminology used before [12].…”

Section: Lagrangian Simulation Of Bimolecular Reaction Controlled By mentioning

confidence: 99%

“…If the growth rate is faster than linear, the transport is anomalous super-diffusion; slower than linear growth rate is sub-diffusion. Super-diffusion is well-known to be described efficiently by the space fractional-order partial differential equation (PDE) [7], and sub-diffusion can be captured by the time fractional PDE [12].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Particle-tracking simulation of fractional diffusion-reaction processes

Zhang

Papelis

2011

Phys. Rev. E

Self Cite

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Abstract. Computer simulation of reactive transport in heterogeneous systems remains a challenge, due to the multi-scale nature of reactive dynamics and the non-Fickian behavior of transport. This study develops a fully Lagrangian approach via particle tracking to describe the reactive transport controlled by the tempered super-or sub-diffusion. In the particle-tracking algorithm, the local-scale reaction is affected by the interaction radius between adjacent reactants, whose motion can be simulated by the Langevin equations corresponding to the tempered stable models. Lagrangian simulation results show that the transient superdiffusion enhances reaction by enhancing the degree of mixing of reactants. The proposed particle-tracking scheme can also be extended conveniently to multi-scaling super-diffusion. For the case of transient sub-diffusion, the trapping of solutes in the immobile phase can either decrease or accelerate the reaction rate, depending on the initial condition of reactant particles. Further practical applications show that the new solver efficiently captures bimolecular reactions observed in laboratories.

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Anomalous solute transport in saturated porous media: Relating transport model parameters to electrical and nuclear magnetic resonance properties

Swanson

Binley

Keating

et al. 2015

Water Resources Research

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The advection-dispersion equation (ADE) fails to describe commonly observed non-Fickian solute transport in saturated porous media, necessitating the use of other models such as the dual-domain mass-transfer (DDMT) model. DDMT model parameters are commonly calibrated via curve fitting, providing little insight into the relation between effective parameters and physical properties of the medium. There is a clear need for material characterization techniques that can provide insight into the geometry and connectedness of pore spaces related to transport model parameters. Here, we consider proton nuclear magnetic resonance (NMR), direct-current (DC) resistivity, and complex conductivity (CC) measurements for this purpose, and assess these methods using glass beads as a control and two different samples of the zeolite clinoptilolite, a material that demonstrates non-Fickian transport due to intragranular porosity. We estimate DDMT parameters via calibration of a transport model to column-scale solute tracer tests, and compare NMR, DC resistivity, CC results, which reveal that grain size alone does not control transport properties and measured geophysical parameters; rather, volume and arrangement of the pore space play important roles. NMR cannot provide estimates of more-mobile and less-mobile pore volumes in the absence of tracer tests because these estimates depend critically on the selection of a material-dependent and flow-dependent cutoff time. Increased electrical connectedness from DC resistivity measurements are associated with greater mobile pore space determined from transport model calibration. CC was hypothesized to be related to length scales of mass transfer, but the CC response is unrelated to DDMT.

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Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications

Cited by 305 publications

References 143 publications

Numerical simulation for two-dimensional Riesz space fractional diffusion equations with a nonlinear reaction term

Numerical simulation for two-dimensional Riesz space fractional diffusion equations with a nonlinear reaction term

Particle-tracking simulation of fractional diffusion-reaction processes

Anomalous solute transport in saturated porous media: Relating transport model parameters to electrical and nuclear magnetic resonance properties

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