Understanding and modeling transport of solutes in porous media is a critical issue in the environmental protection. The common model is the advective–dispersive equation (ADE) describing the superposition of the advective transport and the Brownian motion in water-filled pore space.
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References
Bear J (1972) Dynamics of Fluids in Porous Media. Dove Publications, New York.
Jury WA (1988) Solute transport and dispersion. In: Steffen WL, Denmead OT (eds.), Flow and transport in the Natural Environment: Advances and Applications. Springer, Berlin, pp. 1-16.
Khan AUH, Jury WA (1990) A laboratory test of the dispersion scale effect, J. Contam. Hydrol., 5:119-132.
Porro I, Wierenga PJ, Hills RG (1993) Solute transport through large uniform and layered soil columns, Water Resour. Res., 29:1321-1330.
Snow VO, Clothier BE, Scotter DR, White RE (1994) Solute transport in a layered field soil: experiments and modelling using the convection-dispersion approach, J. Contam. Hydrol., 16:339-358.
Yasuda H, Berndtsson R, Barri A, Jinno K (1994) Plot-scale solute transport in a semiarid agricultural soil, Soil Sci. Soc. Am. J., 58:1052-1060.
Pachepsky Ya, Benson DA, Rawls W (2000) Simulating scale-dependent solute transport in soils with the fractional advective-dispersive equation, Soil Sci. Soc. Am. J., 64:1234-1243.
Zhang R, Huang K, Xiang J (1994) Solute movement through homogeneous and heterogeneous soil columns, Adv. Water Resour., 17:317-324.
Nielsen DR, Van Genuchten MTh, Biggar JW (1986) Water flow and solute transport processes in the unsaturated zone, Water Resour. Res., 22(9, Suppl.): 89S-108S.
Vachaud G, Vauclin M, Addiscott TM (1990) Solute transport in the vadose zone: a review of models. In Proceedings of the International Symposium on Water Quality Modeling of Agricultural Non-Point Sources, Part 1, 19-23 June 1988 pp. 81-104. Logan, UT. USDA-ARS. U.S. Government Printing Office, Washington, DC.
Bhatacharya R, Gupta VK (1990) Application of the central limit theorem to solute transport in saturated porous media: from kinetic to field scales. In: Cushman, JH (eds.), Dynamics of Fluids in Hierarchical Porous Media. Academic Press, New York, pp. 97-124.
Montroll EW, Weiss GH (1965) Random walks on lattices. II, J. Math. Phys., 6:167-183.
Sher H, Lax M (1973a) Stochastic transport in a disordered solid. I. Theory, Phys. Rev. B, 7:4491-4502.
Sher H, Lax M (1973b) Stochastic transport in a disordered solid. II. Impurity conduction, Phys. Rev. B, 7:4502-4519.
Berkowitz B, Klafter J, Metzler, Scher H (2002) Physical pictures of transport in heterogeneous media: advection-dispersion, random-walk, and fractional derivative formulations, Water Resour. Res., 38(10) W1191, doi: 10.1029/2001WR001030.
Metzler R, Klafter J (2004) The restaurant at the end of the random walk: recent developments in the description of anomalous transport by fractional dynamics, J. Phys. A Math. Gen.,37:R161-R208, doi:10.1088/0305-4470/37/31/R01.
Shlesinger MF, Klafter J, Wong YM (1982) Random walks with infinite spatial and temporal moments, J. Stat. Phys., 27:499-512.
Taylor SJ (1986) The measure theory of random fractals, Math. Proc. Camb. Phil. Soc., 100(3):383-406.
Weeks ER, Solomon TH, Urbach JS, Swinney HL (1995) Observation of anomalous diffusion and Lévy flights. In: Shlesinger MF, Zaslavsky GM, Frisch U (eds.), Lévy Flights and Related Topics in Physics. Springer, New York, pp. 51-71.
Benson DA(1998) The fractional advection-dispersion equation: development and application, PhD Thesis, University of Nevada, Reno.
Benson DA, Wheatcraft SW, Meerschaert MM (2000) Application of a fractional advection-dispersion equation, Water Resour. Res., 36(6):1403-1412.
Zaslavsky GM (1994) Renormalization group theory of anomalous transport in systems with Hamiltonian chaos, Chaos, 4(1):25-33.
Compte A (1996) Stochastic foundations of fractional dynamics, Phys. Rev. E, 53(4):4191-4193.
Compte A (1997) Continuous time random walks on moving fluids, Phys. Rev. E, 55(6):6821-6831.
Saichev AI, Zaslavsky GM (1997) Fractional kinetic equations: solutions and applications, Chaos, 7:753-764.
Chaves AS (1998) A fractional diffusion equation to describe Lévy flights, Phys. Lett. A, 239:13-16.
Metzler R, Klafter J, Sokolov IM (1998) Anomalous transport in external fields: continuous time ransom walks and fractional diffusion equations extended, Phys. Rev. E, 58:1621-1633.
Benson DA, Schumer R, Meerschaert MM, Wheatcraft SW (2001) Fractional dispersion, Levy motion, and the MADE tracer tests, Transp. Porous Media, 42:211-240.
Lu S, Molz FJ, Fix GJ (2002) Possible problems of scale dependency in applications of the three-dimensional fractional advection-dispersion equation to natural porous media, Water Resour. Res. 38(9):1165, doi:10.1029/ 2001WR000624.
Meerschaert MM, Benson DA, Bäumer B (1999) Multidimensional advection and fractional dispersion, Phys. Rev. E, 59(5):5026-5028.
Meerschaert MM, Benson DA, Bäumer B (2001) Operator Levy motion and multiscaling anomalous diffusion, Phys. Rev. E, 63(2):1112-1117.
Zhou LZ, Selim HM (2003) Application of the fractional advection- dispersion equation in porous media. Soil Sci. Soc. Am. J., 67(4):1079-1084.
Deng Z-Q, Singh VP, Bengtsson L (2004) Numerical solution of fractional advection-dispersion equation, ASCE J. Hydraulic Eng., 130(5):422-431.
Zhang X, Crawford JW, Deeks LK, Stutter MI, Bengough AG, Young IM (2005) A mass balance based numerical method for the fractional advectiondispersion equation: theory and application, Water Resour. Res., 41, W07029, doi: 10.1029/2004WR003818.
Lynch VE, Carreras BA, del-Castillo-Negrete D, Ferreiras-Mejias KM, Hicks HR (2003) Numerical method for the solution of partial differential equations with fractional order. J. Comp. Phys., 192:406-421.
Oldham KB, Spanier J (1974) The Fractional Calculus. Academic Press, New York.
Liu F, Anh V, Turner I (2004) Numerical solution of the space fractional Fokker-Planck equation, J. Comp. Appl. Math., 166:209-219.
Meerschaert MM, Tadjeran C (2004) Finite difference approximation for fractional advection-dispersion flow equations, J. Com. Appl. Math., 172:65-77.
Meerschaert MM, Tadjeran C (2006) Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Num. Math., 56:80-90.
van Genuchten MTh, Parker JC (1984) Boundary conditions for displacement experiments through short soil columns, Soil Sci. Soc Am J., 48:703-708.
San Jose MF, Pachepsky Ya, Rawls W (2005) Solute transport simulated with the fractional advective-dispersive equation. In: Agrawal O, Tenreiro Machado JA, Sabatier J. (eds.), Fractional Derivatives and Their Applications. IDECT/CIE2005. ISBN: 0-7918-3766-1.
Samko SG, Kilbas AA, Marichev OI (1993). Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science, New York.
Toride N, Leij FJ, van Genuchten MTh (1995) The CXTFIT code for estimating transport parameters from laboratory or field tracer experiments. Version 2.0. Research Report 137, US Salinity Lab, Riverside, CA.
Press WH, Teukolsky SA, Vetterling WT, Flannery, BP (1992) Numerical Recipes in FORTRAN 77. The Art of Computing, 2nd edition. Cambridge University Press, New York.
Dyson JS, White RE (1987) A comparison of the convective-dispersive equation and transfer function model for predicting chloride leaching through an undisturbed structured clay soil, J. Soil Sci., 38:157-172.
van Genuchten MTh, Wierenga PJ (1976) Mass transfer in sorbing porous media. I. Analytical solutions, Soil Sci. Soc. Am. J., 40:473-481.
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Martinez, F.S.J., Pachepsky, Y.A., Rawls, W.J. (2007). Fractional Advective-Dispersive Equation as a Model of Solute Transport in Porous Media. In: Sabatier, J., Agrawal, O.P., Machado, J.A.T. (eds) Advances in Fractional Calculus. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-6042-7_14
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