Skip to main content
SpringerLink
Log in

Parameter estimation for fractional dispersion model for rivers

  • Original Article
  • Published:
Environmental Fluid Mechanics Aims and scope Submit manuscript

Abstract

The fractional dispersion model for natural rivers, extended by including a first order reaction term, contains four parameters. In order to estimate these parameters a fractional Laplace transform-based method is developed in this paper. Based on 76 dye test data measured in natural streams, the new parameter estimation method shows that the fractional dispersion operator parameter F is the controlling parameter causing the non-Fickian dispersion and F does not take on an integer constant of 2 but instead varies in the range of 1.4–2.0. The adequacy of the fractional Laplace transform-based parameter estimation method is determined by computing dispersion characteristics of the extended fractional dispersion model and these characteristics are compared with those observed from 12 dye tests conducted on the US rivers, including Mississippi, Red, and Monocacy. The agreement between computed and observed dispersion characteristics is found to be good. When combined with the fractional Laplace transform-based parameter estimation method, the extended fractional dispersion model is capable of accurately simulating the non-Fickian dispersion process in natural streams.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Bencala KE, Walters RA (1983) Simulation of solute transport in a mountain pool-and-riffle stream: a transient storage model. Water Resources Research 19(3):718–724

    Article  CAS  Google Scholar 

  • Bakunin OG (2004) Correlation effects and turbulent diffusion scalings. Rep Prog Phys 67:965–1032

    Article  Google Scholar 

  • Berkowitz B, Scher H (1995). On characterization of anomalous dispersion in porous and fractured media Water Resourc Res 31(6):1461–1466

    Google Scholar 

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

    Article  Google Scholar 

  • Chatwin PC (1973) A calculation illustrating effects of the viscous sub-layer on longitudinal dispersion. Quart J Mech Appl Math 26:427–439

    Google Scholar 

  • Chaves AS (1998) A fractional diffusion equation to describe Lévy flights. Phys Lett A 239(1–2):13–16

    Article  CAS  Google Scholar 

  • Clarke DD, Meerschaert MM, Wheatcraft SW (2005) Fractal travel time estimates for dispersive contaminants. Ground Water 43(3):401–407

    Article  CAS  Google Scholar 

  • Davis PM, Atkinson TC, Wigley TML (2000) Longitudinal dispersion in natural channels: 2. The roles of shear flow dispersion and dead zones in the River Sever, UK. Hydrol Earth Syst Sci 4(3):355–371

    Article  Google Scholar 

  • Day TJ (1975) Longitudinal dispersion in natural channels. Water Resourc Res 11(6):909–918

    Google Scholar 

  • Day TJ, Wood IR (1976) Similarity of the mean motion of fluid particles dispersing in a natural channel. Water Resourc Res 12(4):655–666

    Google Scholar 

  • Debnath L (1995) Integral transforms and their applications. CRC Press, USA, pp. 11–12 and pp. 98–117

  • Del-Castillo-Negrete D, Carreras BA, Lynch VE (2003) Front dynamics in reaction-diffusion systems with Levy flights: A fractional diffusion approach. Phys Revi Lett 91(1):018302

    Article  CAS  Google Scholar 

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

    Article  Google Scholar 

  • Ding D, Liu PL-F (1989) An operator-splitting algorithm for two-dimensional convection-dispersion-reaction problems. Int J Numer Methods Eng 28:1023–1040

    Article  Google Scholar 

  • Di Toro DM (2001) Sediment flux modeling. J Wiley, New York, USA pp. 3–55

    Google Scholar 

  • Falconer RA, Liu S (1988) Modeling solute transport using QUICK scheme. J Environ Eng 114(1): 3–20

    Article  CAS  Google Scholar 

  • Fix GJ, Roop JP (2004) Least squares finite element solution of a fractional order two-point boundary value problem. Comput Math. Appl 48:1017–1033

    Article  Google Scholar 

  • Fischer HB, List EJ, Koh RCY, Imberger J, Brooks NH (1979) Mixing in inland and coastal waters. Academic Press, New York, USA pp. 30–138

    Google Scholar 

  • Godfrey RG, Fredrick BJ (1970) Stream dispersion at selected sites. Professional Paper 433-K, US Geological Survey

  • Harris JW, Stocker H (1998) Handbook of mathematics and computational science. Springer-Verlag, New York, pp. 33–36, 535–537, 736–764

  • Holly FM, Preissmann A (1977) Accurate calculation of transport in two-dimensions. J Hydraulics Division 103(11):1259–1277

    Google Scholar 

  • Holly FM, Usseglio-Polatera J-D (1984) Accurate two-dimensional simulation of advective-diffusive-reactive transport. J Hydraulic Eng 127(9):728–737

    Google Scholar 

  • Hunt B (1999) Dispersion model for mountain streams. J Hydraulic Eng 125(2):99–105

    Article  Google Scholar 

  • Karpik SR, Crockett SR (1997) Semi-Lagrangian algorithm for two-dimensional advection-diffusion equation on curvilinear coordinate meshes. J Hydraulic Eng 123(5):389–401

    Article  Google Scholar 

  • Komatsu T, Ohgushi K, Asai K (1997) Refined numerical scheme for advective transport in diffusion simulation. J Hydraulic Eng 123(1):41–50

    Article  Google Scholar 

  • Langlands TAM, Henry BI (2005) The accuracy and stability of an implicit solution method for the fractional diffusion equation. J Comput Phys 205(2):719–736

    Article  Google Scholar 

  • Lin B, Falconer RA (1997) Tidal flow and transport modeling using ULTIMATE QUICKEST scheme. J Hydraulic Eng 123(4):303–314

    Article  Google Scholar 

  • Lynch VE, Carreras BA, del-Castillo-Negrete D, Ferreira-Mejias KM, Hicks HR (2003) Numerical methods for the solution of partial differential equations of fractional order. J Comput Phys 192:406–421

    Article  Google Scholar 

  • Meerschaert MM, Mortensen J, Wheatcraft SW (2006) Fractional vector calculus for fractional dispersion. Phys A: Stat Mech Appl 367:181–190

    Article  Google Scholar 

  • Meerschaert MM, Tadjeran C (2003) Finite difference approximations for fractional dispersion flow equations. J Comput Appl Math 172:65–77

    Article  Google Scholar 

  • Miller KS, Ross B (1993) An introduction to the fractional calculus and fractional differential. J Wiley, New York, pp. 21–185

    Google Scholar 

  • Nordin CF, Sabol GV (1974) Empirical data on longitudinal dispersion in rivers. Water Resources Investigations 20–74 US Geological Survey

  • Nordin CF, Troutman BM (1980) Longitudinal dispersion in rivers: the persistence of skewness in observed data. Water Resourc Res 16(1):123–128

    Google Scholar 

  • Oldham KB, Spanier J (1974) The fractional calculus. Academic Press, New York, pp. 45–134

    Google Scholar 

  • Podlubny I (1999) Fractional differential equations. Academic Press, San Diego, pp. 41–242

    Google Scholar 

  • Press WH, Flannery BP, Teukolsky SA, Vetterling WT (1988) Numerical recipes. Cambridge University Press, New York, pp. 77–101, 615–666

  • Runkel RL (1998) One-dimensional transport with inflow and storage (OTIS): A solute transport model for streams and rivers. Water Resources Investigations Report 98–4018, US Geological Survey, Denver

  • Schmid BH (2003) Temporal moments routing in streams and rivers with transient storage. Adv Water Resourc 26(9):1021–1027

    Article  Google Scholar 

  • Schumer R, Benson DA, Meerschaert MM, Baeumer B (2003) Fractal mobile/immobile solute transport. Water Resourc Res 39(10):1296:doi:10.1029/2003WR002141

    Article  Google Scholar 

  • Singh VP (1998) Entropy-based parameter estimation in hydrology. Kluwer Academic Publishers, Netherlands, pp. 12–39

    Google Scholar 

  • Stefanovic DL, Stefan HD (2001) Dispersion simulation in two-dimensional tidal flow. J Hydraulic Eng 110(7):905–926

    Google Scholar 

  • Sullivan PJ (1971) Longitudinal dispersion within a two-dimensional turbulent shear flow. J Fluid Mech 49:551–576

    Article  Google Scholar 

  • Wheatcraft SW, Tyler S (1988) An explanation of scale dependent dispersivity in heterogeneous aquifers using concepts of fractal geometry. Water Resourc Res 24(4):566–578

    CAS  Google Scholar 

  • Wörman A (1998) Analytical solution and timescale for transport of reacting solutes in rivers and streams. Water Resourc Res 34(10):2703–2716

    Article  Google Scholar 

  • Yotsukura N, Fischer HB, Sayre WW (1970) Measurement of mixing characteristics of the Missouri River between Sioux City, Iowa, and Plattsmouth, Nebraska. Water-Supply Paper 1899-G, US Geological Survey

Download references

Author information

Authors and Affiliations

  1. Department of Civil and Environmental Engineering, Louisiana State University, Baton Rouge, LA, 70803-6405, USA

    Zhiqiang Deng

  2. Department of Water Resources Engineering, Lund University, Lund, Sweden

    Lars Bengtsson

  3. Department of Biological and Agricultural Engineering, Texas A and M University, College Station, Texas, 77843-2117, USA

    Vijay P. Singh

Authors
  1. Zhiqiang Deng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lars Bengtsson
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Vijay P. Singh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhiqiang Deng.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Deng, Z., Bengtsson, L. & Singh, V.P. Parameter estimation for fractional dispersion model for rivers. Environ Fluid Mech 6, 451–475 (2006). https://doi.org/10.1007/s10652-006-9004-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10652-006-9004-5

Keywords

Search

Navigation