Skip to main content
SpringerLink
Log in

Fractional Advective-Dispersive Equation as a Model of Solute Transport in Porous Media

  • Chapter
Book cover Advances in Fractional Calculus

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.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 169.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 219.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 219.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bear J (1972) Dynamics of Fluids in Porous Media. Dove Publications, New York.

    Google Scholar 

  2. 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.

    Google Scholar 

  3. Khan AUH, Jury WA (1990) A laboratory test of the dispersion scale effect, J. Contam. Hydrol., 5:119-132.

    Article  Google Scholar 

  4. Porro I, Wierenga PJ, Hills RG (1993) Solute transport through large uniform and layered soil columns, Water Resour. Res., 29:1321-1330.

    Article  Google Scholar 

  5. 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.

    Article  Google Scholar 

  6. 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.

    Google Scholar 

  7. 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.

    Google Scholar 

  8. Zhang R, Huang K, Xiang J (1994) Solute movement through homogeneous and heterogeneous soil columns, Adv. Water Resour., 17:317-324.

    Article  Google Scholar 

  9.  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.

    Google Scholar 

  10.  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.

    Google Scholar 

  11. 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.

    Google Scholar 

  12. Montroll EW, Weiss GH (1965) Random walks on lattices. II, J. Math. Phys., 6:167-183.

    Article  MathSciNet  Google Scholar 

  13. Sher H, Lax M (1973a) Stochastic transport in a disordered solid. I. Theory, Phys. Rev. B, 7:4491-4502.

    Article  Google Scholar 

  14. Sher H, Lax M (1973b) Stochastic transport in a disordered solid. II. Impurity conduction, Phys. Rev. B, 7:4502-4519.

    Article  Google Scholar 

  15.  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.

    Google Scholar 

  16. 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.

    Google Scholar 

  17. Shlesinger MF, Klafter J, Wong YM (1982) Random walks with infinite spatial and temporal moments, J. Stat. Phys., 27:499-512.

    Article  MATH  MathSciNet  Google Scholar 

  18. Taylor SJ (1986) The measure theory of random fractals, Math. Proc. Camb. Phil. Soc., 100(3):383-406.

    Article  MATH  Google Scholar 

  19. 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.

    Chapter  Google Scholar 

  20. Benson DA(1998) The fractional advection-dispersion equation: development and application, PhD Thesis, University of Nevada, Reno.

    Google Scholar 

  21. Benson DA, Wheatcraft SW, Meerschaert MM (2000) Application of a fractional advection-dispersion equation, Water Resour. Res., 36(6):1403-1412.

    Article  Google Scholar 

  22. Zaslavsky GM (1994) Renormalization group theory of anomalous transport in systems with Hamiltonian chaos, Chaos, 4(1):25-33.

    Article  MATH  MathSciNet  Google Scholar 

  23. Compte A (1996) Stochastic foundations of fractional dynamics, Phys. Rev. E, 53(4):4191-4193.

    Article  Google Scholar 

  24. Compte A (1997) Continuous time random walks on moving fluids, Phys. Rev. E, 55(6):6821-6831.

    Article  Google Scholar 

  25. Saichev AI, Zaslavsky GM (1997) Fractional kinetic equations: solutions and applications, Chaos, 7:753-764.

    Article  MATH  MathSciNet  Google Scholar 

  26. Chaves AS (1998) A fractional diffusion equation to describe Lévy flights, Phys. Lett. A, 239:13-16.

    Article  MATH  MathSciNet  Google Scholar 

  27.  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.

    Article  Google Scholar 

  28. Benson DA, Schumer R, Meerschaert MM, Wheatcraft SW (2001) Fractional dispersion, Levy motion, and the MADE tracer tests, Transp. Porous Media, 42:211-240.

    Article  MathSciNet  Google Scholar 

  29.  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.

    Google Scholar 

  30. Meerschaert MM, Benson DA, Bäumer B (1999) Multidimensional advection and fractional dispersion, Phys. Rev. E, 59(5):5026-5028.

    Article  Google Scholar 

  31. Meerschaert MM, Benson DA, Bäumer B (2001) Operator Levy motion and multiscaling anomalous diffusion, Phys. Rev. E, 63(2):1112-1117.

    Article  Google Scholar 

  32. Zhou LZ, Selim HM (2003) Application of the fractional advection- dispersion equation in porous media. Soil Sci. Soc. Am. J., 67(4):1079-1084.

    Google Scholar 

  33. Deng Z-Q, Singh VP, Bengtsson L (2004) Numerical solution of fractional advection-dispersion equation, ASCE J. Hydraulic Eng., 130(5):422-431.

    Article  Google Scholar 

  34. 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.

    Article  Google Scholar 

  35. 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.

    Article  MATH  Google Scholar 

  36. Oldham KB, Spanier J (1974) The Fractional Calculus. Academic Press, New York.

    MATH  Google Scholar 

  37. Liu F, Anh V, Turner I (2004) Numerical solution of the space fractional Fokker-Planck equation, J. Comp. Appl. Math., 166:209-219.

    Article  MATH  MathSciNet  Google Scholar 

  38. Meerschaert MM, Tadjeran C (2004) Finite difference approximation for fractional advection-dispersion flow equations, J. Com. Appl. Math., 172:65-77.

    Article  MATH  MathSciNet  Google Scholar 

  39. Meerschaert MM, Tadjeran C (2006) Finite difference approximations for two-sided space-fractional partial differential equations, Appl. Num. Math., 56:80-90.

    Article  MATH  MathSciNet  Google Scholar 

  40. van Genuchten MTh, Parker JC (1984) Boundary conditions for displacement experiments through short soil columns, Soil Sci. Soc Am J., 48:703-708.

    Google Scholar 

  41.  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.

    Google Scholar 

  42. Samko SG, Kilbas AA, Marichev OI (1993). Fractional Integrals and Derivatives: Theory and Applications. Gordon and Breach Science, New York.

    MATH  Google Scholar 

  43. 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.

    Google Scholar 

  44. 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.

    Google Scholar 

  45. 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.

    Article  Google Scholar 

  46. van Genuchten MTh, Wierenga PJ (1976) Mass transfer in sorbing porous media. I. Analytical solutions, Soil Sci. Soc. Am. J., 40:473-481.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department de Matematica Aplicada, ETSIA-UPM, Avd. de la Complutense s/n, 28040, Madrid, Spain

    F. San Jose Martinez

  2. Environmental Microbial Safety Laboratory, USDA-ARS-BA-ANRI-EMSL, MD 20705, Beltsville

    Y. A. Pachepsky

  3. Hydrology and Remote Sensing Laboratory, USDA-ARS-BA-ANRI-RSL, MD 20705, Beltsville

    W. J. Rawls

Authors
  1. F. San Jose Martinez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Y. A. Pachepsky
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. W. J. Rawls
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Université Bordeaux 1, – ENSEIRB, 351 cours de la Libération, F33405, Talence Cedex, France

    Jocelyn Sabatier

  2. Southern Illinois University, 62901, Carbondale, IL, USA

    Om Prakash Agrawal

  3. Institute of Engineering of Porto, Rua Dr. António Bernardino de Almeida, 4200-072, Porto, Portugal

    J. A. Tenreiro Machado

Rights and permissions

Reprints and permissions

Copyright information

© 2007 Springer

About this chapter

Cite this chapter

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

Download citation

  • DOI: https://doi.org/10.1007/978-1-4020-6042-7_14

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-1-4020-6041-0

  • Online ISBN: 978-1-4020-6042-7

  • eBook Packages: EngineeringEngineering (R0)

Publish with us

Policies and ethics

Search

Navigation