Eulerian derivation of the fractional advection–dispersion equation

Journal of Contaminant Hydrology - Tập 48 - Trang 69-88 - 2001
Rina Schumer1, David A. Benson1, Mark M. Meerschaert2, Stephen W. Wheatcraft3
1Division of Hydrologic Sciences, Desert Research Institute, 2215 Raggio Parkway, Reno, NV 89512, USA
2Department of Mathematics, University of Nevada, Reno, NV 89577, USA
3Department of Geological Sciences, University of Nevada, Reno, NV 89577, USA

Tài liệu tham khảo

Bear, 1961, Some experiments on dispersion, J. Geophys. Res., 66, 2455, 10.1029/JZ066i008p02455 Bear, 1972 Benson, D.A., 1998. The fractional advection–dispersion equation: development and application. Unpublished PhD dissertation, University of Nevada. Benson, 2000, Fractional dispersion, Lévy motion, and the MADE tracer tests, Transp. Porous Media Benson, 2000, Application of a fractional advection–dispersion equation, Water Resour. Res., 36, 1403, 10.1029/2000WR900031 Berkowitz, 1995, On characterization of anomalous dispersion in porous and fractured media, Water Resour. Res., 31, 1461, 10.1029/95WR00483 Berkowitz, 1998, Theory of anomalous chemical transport in random fracture networks, Phys. Rev. E, 57, 5858, 10.1103/PhysRevE.57.5858 Bhattacharya, 1990, Application of central limit theorems to solute transport in saturated porous media: from kinetic to field scales Blank, L., 1996. Numerical treatment of differential equations of fractional order, Manchester Centre for Computational Mathematics Numerical Analysis Report No. 287. Brusseau, 1992, Transport of rate-limited sorbing solutes in heterogeneous porous media: application of a one-dimensional multifactor nonideality model to field data, Water Resour. Res., 28, 2485, 10.1029/92WR00907 Chaves, 1998, A fractional diffusion equation to describe Lévy flights, Phys. Lett. A, 239, 13, 10.1016/S0375-9601(97)00947-X Compte, 1996, Stochastic foundations of fractional dynamics, Phys. Rev. E, 53, 4191, 10.1103/PhysRevE.53.4191 Crank, 1976 Cushman, 1993, Fractal dispersion and diffusion: preface, Transp. Porous Media, 13, 1, 10.1007/BF00613268 Cushman, 1993, Nonlocal dispersion in media with continuously evolving scales of heterogeneity, Transp. Porous Media, 13, 123, 10.1007/BF00613273 Cushman, J.H., Ginn, T.R., in review. The fractional ADE is a classical mass balance with convolution-Fickian flux. Water Resour. Res. Dagan, 1984, Solute transport in heterogeneous porous formations, J. Fluid Mech., 145, 151, 10.1017/S0022112084002858 Dagan, 1990, Transport in heterogeneous formations: spatial moments, ergodicity and effective dispersion, Water Resour. Res., 26, 1281, 10.1029/WR026i006p01281 Dagan, 1991, Dispersion of a passive solute in non-ergodic transport by steady velocity fields in heterogeneous formations, J. Fluid Mech., 233, 197, 10.1017/S0022112091000459 Feller, 1968, vol. I Feller, 1971, vol. II Freeze, 1979 Gelhar, 1993 Gelhar, 1983, Three-dimensional stochastic analysis of macrodispersion in aquifers, Water Resour. Res., 19, 161, 10.1029/WR019i001p00161 Gnedenko, 1968 Gorenflo, 1998, Fractional calculus and stable probability distributions, Arch. Mech., 50, 377 Gorenflo, 1998, Random walk models for space-fractional diffusion processes, Fractional Calculus Appl. Anal., 1, 167 Haggerty, 1995, Multiple-rate mass transfer for modeling diffusion and surface reactions in media with pore-scale heterogeneity, Water Resour. Res., 31, 2383, 10.1029/95WR10583 Kochubei, 1990, Fractional-order diffusion, Differ. Equations, 26, 485 Liu, 1996, Discrimination of fractional Brownian motion and fractional Gaussian noise structures in permeability and related property distributions with range analyses, Water Resour. Res., 32, 2601, 10.1029/96WR01394 Liu, 1997, Multifractal analyses of hydraulic conductivity distributions, Water Resour. Res., 33, 2483, 10.1029/97WR02188 Makse, 1998, Tracer dispersion in porous media with spatial correlations, Phys. Rev. Lett. Meerschaert, 1999, Multidimensional advection and fractional dispersion, Phys. Rev. E, 59, 5026, 10.1103/PhysRevE.59.5026 Metzler, 1994, Fractional model equation for anomalous diffusion, Physica A, 211, 13, 10.1016/0378-4371(94)90064-7 Miller, 1993 Molz, 1997, Fractional Brownian motion and fractional Gaussian noise in subsurface hydrology: a review, presentation of fundamental properties and extensions, Water Resour. Res., 33, 2273, 10.1029/97WR01982 Molz, 1999, An examination of preferential flow structure in stochastic fractal models of hydraulic conductivity distributions, GSA Abstr. Program, 317, A150 Montroll, 1965, Random walks on lattices: II, J. Math Phys., 6, 167, 10.1063/1.1704269 Neuman, 1993, Eulerian–Lagrangian theory of transport in space-time nonstationary velocity fields: exact nonlocal formalism by conditional moments and weak approximation, Water Resour. Res., 29, 633, 10.1029/92WR02306 Oldham, 1974 Osler, 1971, Taylor's series generalized for fractional derivatives and applications, SIAM J. Math. Anal., 2, 37, 10.1137/0502004 Painter, 1996, Stochastic interpolation of aquifer properties using fractional Lévy motion, Water Resour. Res., 32, 1323, 10.1029/95WR03485 Pollock, D.W., 1994. User's guide for MODPATH/MODPATH-PLOT, Version 3: a particle tracing post-processing package for MODFLOW, the United States Geological Survey finite difference groundwater model. U.S. Geological Survey, Open-File Report 94-464, Reston, VA. Roman, 1992, Fractional diffusion equation on fractals: three-dimensional case and scattering function, J. Phys. A: Math. Gen., 25, 2107, 10.1088/0305-4470/25/8/024 Ross, 1997 Rubin, 1997, Transport of inert solutes by groundwater: recent developments and current issues, 115 Sahimi, 1993, Fractal and superdiffusive transport and hydrodynamic dispersion in heterogeneous porous media, Transp. Porous Media, 13, 3, 10.1007/BF00613269 Samko, 1993 Scher, 1973, Stochastic transport in a disordered solid: I. Theory, Phys. Rev. B, 7, 4491, 10.1103/PhysRevB.7.4491 Serrano, 1995, Forecasting scale-dependent dispersion from spills in heterogeneous aquifers, J. Hydrol., 169, 151, 10.1016/0022-1694(94)02663-V Tompkins, 1994, Prediction of solute dispersion in heterogeneous porous media: effects of ergodicity and hydraulic conductivity discretization, J. Hydrol., 159, 105, 10.1016/0022-1694(94)90251-8 Tsallis, 1995, Statistical–mechanical foundation of the ubiquity of Lévy Distributions in nature, Phys. Rev. Lett., 75, 3589, 10.1103/PhysRevLett.75.3589 Zaslavsky, 1994, Renormalization group theory of anomalous transport in systems with Hamiltonian chaos, Chaos, 4, 25, 10.1063/1.166054 Zhan, 1999, On the ergodicity hypothesis in heterogeneous formations, Math. Geol., 31, 113 Zheng, 1998, Numerical simulation of tracer tests in heterogeneous aquifer, J. Environ. Eng., 124, 510, 10.1061/(ASCE)0733-9372(1998)124:6(510) Zumofen, 1990, Anomalous transport: a one-dimensional stochastic model, Chem. Phys., 146, 433, 10.1016/0301-0104(90)80062-3