Use of a variable-index fractional-derivative model to capture transient dispersion in heterogeneous media.

[1]  G. Taylor.  Conditions under which dispersion of a solute in a stream of solvent can be used to measure molecular diffusion , 1954, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[2]  H. Wold,et al.  Limit Distributions for Sums of Independent Variables , 1955 .

[3]  John H. Cushman,et al.  Development of stochastic partial differential equations for subsurface hydrology , 1987 .

[4]  Peter K. Kitanidis,et al.  Prediction by the method of moments of transport in a heterogeneous formation , 1988 .

[5]  E. Eric Adams,et al.  Field study of dispersion in a heterogeneous aquifer: 1. Overview and site description , 1992 .

[6]  B. Ross,et al.  Integration and differentiation to a variable fractional order , 1993 .

[7]  S. Gorelick,et al.  Multiple‐Rate Mass Transfer for Modeling Diffusion and Surface Reactions in Media with Pore‐Scale Heterogeneity , 1995 .

[8]  M. Taqqu,et al.  Stable Non-Gaussian Random Processes : Stochastic Models with Infinite Variance , 1995 .

[9]  J. Hadermann,et al.  The Grimsel (Switzerland) migration experiment : integrating field experiments, laboratory investigations and modelling , 1996 .

[10]  Paul S. Addison,et al.  A Particle Tracking Model for Non-Fickian Subsurface Diffusion , 1998 .

[11]  M. Dentz,et al.  Temporal behaviour of a solute cloud in a chemically heterogeneous porous medium , 1999, Journal of Fluid Mechanics.

[12]  Sean Andrew McKenna,et al.  On the late‐time behavior of tracer test breakthrough curves , 2000 .

[13]  C. Zheng,et al.  A dual‐domain mass transfer approach for modeling solute transport in heterogeneous aquifers: Application to the Macrodispersion Experiment (MADE) site , 2000 .

[14]  V. Gonchar,et al.  Self and spurious multi-affinity of ordinary Levy motion, and pseudo-Gaussian relations , 2000 .

[15]  M. Meerschaert,et al.  Limit Distributions for Sums of Independent Random Vectors: Heavy Tails in Theory and Practice , 2001 .

[16]  H. Scher,et al.  The Role of Probabilistic Approaches to Transport Theory in Heterogeneous Media , 2001 .

[17]  D. Benson,et al.  Eulerian derivation of the fractional advection-dispersion equation. , 2001, Journal of contaminant hydrology.

[18]  D. Benson,et al.  Fractional Dispersion, Lévy Motion, and the MADE Tracer Tests , 2001 .

[19]  Carl F. Lorenzo,et al.  Variable Order and Distributed Order Fractional Operators , 2002 .

[20]  Fred J. Molz,et al.  Possible problems of scale dependency in applications of the three‐dimensional fractional advection‐dispersion equation to natural porous media , 2002 .

[21]  D. Benson,et al.  Hydraulic conductivity, velocity, and the order of the fractional dispersion derivative in a highly heterogeneous system , 2002 .

[22]  Raisa E. Feldman,et al.  Limit Distributions for Sums of Independent Random Vectors , 2002 .

[23]  D. Veneziano,et al.  Flow through porous media with multifractal hydraulic conductivity , 2003 .

[24]  B. Berkowitz,et al.  Measurement and analysis of non-Fickian dispersion in heterogeneous porous media. , 2003, Journal of contaminant hydrology.

[25]  Rina Schumer,et al.  Multiscaling fractional advection‐dispersion equations and their solutions , 2003 .

[26]  Carlos F.M. Coimbra,et al.  Mechanics with variable‐order differential operators , 2003 .

[27]  Georg Kosakowski,et al.  Anomalous transport of colloids and solutes in a shear zone. , 2004, Journal of contaminant hydrology.

[28]  Harihar Rajaram,et al.  Stochastic fractal‐based models of heterogeneity in subsurface hydrology: Origins, applications, limitations, and future research questions , 2004 .

[29]  Igor M. Sokolov,et al.  Fractional diffusion in inhomogeneous media , 2005 .

[30]  Guanhua Huang,et al.  Evidence of one-dimensional scale-dependent fractional advection-dispersion. , 2006, Journal of contaminant hydrology.

[31]  Brian Berkowitz,et al.  Non‐Fickian transport and multiple‐rate mass transfer in porous media , 2008 .

[32]  M. Meerschaert,et al.  Heavy‐tailed log hydraulic conductivity distributions imply heavy‐tailed log velocity distributions , 2006 .

[33]  G. Fogg,et al.  Diffusive fractionation of 3H and 3He in groundwater and its impact on groundwater age estimates , 2006 .

[34]  M. Dentz,et al.  Modeling non‐Fickian transport in geological formations as a continuous time random walk , 2006 .

[35]  David A. Benson,et al.  Space-fractional advection-dispersion equations with variable parameters : Diverse formulas , numerical solutions , and application to the MADE-site data , 2007 .

[36]  M. Meerschaert,et al.  Tempered anomalous diffusion in heterogeneous systems , 2008 .

[37]  Donald M. Reeves,et al.  Transport of conservative solutes in simulated fracture networks: 2. Ensemble solute transport and the correspondence to operator‐stable limit distributions , 2008 .

[38]  Boris Baeumer,et al.  Particle tracking for time-fractional diffusion. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[39]  Daniel M. Tartakovsky,et al.  Perspective on theories of non-Fickian transport in heterogeneous media , 2009 .

[40]  D. Benson,et al.  Time and space nonlocalities underlying fractional-derivative models: Distinction and literature review of field applications , 2009 .

[41]  Yong Zhang,et al.  Monte Carlo simulation of superdiffusion and subdiffusion in macroscopically heterogeneous media , 2009 .

[42]  Fawang Liu,et al.  Numerical Methods for the Variable-Order Fractional Advection-Diffusion Equation with a Nonlinear Source Term , 2009, SIAM J. Numer. Anal..

[43]  Marco Dentz,et al.  Effective transport dynamics in porous media with heterogeneous retardation properties , 2009 .

[44]  Y. Chen,et al.  Variable-order fractional differential operators in anomalous diffusion modeling , 2009 .

[45]  S. P. Neuman.  Apparent/spurious multifractality of data sampled from fractional Brownian/Lévy motions , 2009 .

[46]  Chae Young Lim,et al.  Parameter estimation for fractional transport: A particle‐tracking approach , 2009 .

[47]  Mark M. Meerschaert,et al.  Tempered stable Lévy motion and transient super-diffusion , 2010, J. Comput. Appl. Math..

[48]  Donald M. Reeves,et al.  A tempered multiscaling stable model to simulate transport in regional-scale fractured media , 2010 .

[49]  Reply to comment by V. P. Shkilev on “Non‐Fickian transport and multiple‐rate mass transfer in porous media” , 2010 .

[50]  M. Sivapalan,et al.  A subordinated kinematic wave equation for heavy-tailed flow responses from heterogeneous hillslopes , 2010 .

[51]  M. Dentz,et al.  Distribution- versus correlation-induced anomalous transport in quenched random velocity fields. , 2010, Physical review letters.

[52]  W. Chen,et al.  A comparative study of constant-order and variable-order fractional models in characterizing memory property of systems , 2011 .

[53]  Mark M. Meerschaert,et al.  Gaussian setting time for solute transport in fluvial systems , 2011 .

[54]  Alexandre M. Tartakovsky,et al.  Effective pore‐scale dispersion upscaling with a correlated continuous time random walk approach , 2011 .

[55]  B. Martin.  PARAMETER ESTIMATION , 2012, Statistical Methods for Biomedical Research.

[56]  Hongguang Sun,et al.  Finite difference Schemes for Variable-Order Time fractional Diffusion equation , 2012, Int. J. Bifurc. Chaos.

[57]  Alexandre M Tartakovsky,et al.  Flow intermittency, dispersion, and correlated continuous time random walks in porous media. , 2013, Physical review letters.

[58]  Yong Zhang,et al.  Challenges in the Application of Fractional Derivative Models in Capturing Solute Transport in Porous Media: Darcy-Scale Fractional Dispersion and the Influence of Medium Properties , 2013 .

[59]  A. Wyłomańska,et al.  Tempered stable Lévy motion driven by stable subordinator , 2013 .

[60]  T. Ginn,et al.  Using groundwater age distributions to estimate the effective parameters of Fickian and non-Fickian models of solute transport. , 2013, Advances in water resources.

[61]  Graham E. Fogg,et al.  The impact of medium architecture of alluvial settings on non-Fickian transport , 2013 .