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.
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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
Bakunin OG (2004) Correlation effects and turbulent diffusion scalings. Rep Prog Phys 67:965–1032
Berkowitz B, Scher H (1995). On characterization of anomalous dispersion in porous and fractured media Water Resourc Res 31(6):1461–1466
Benson DA, Wheatcraft SW, Meerschaert MM (2000) Application of a fractional advection-dispersion equation. Water Resourc Res 36(6):1403–1412
Chatwin PC (1973) A calculation illustrating effects of the viscous sub-layer on longitudinal dispersion. Quart J Mech Appl Math 26:427–439
Chaves AS (1998) A fractional diffusion equation to describe Lévy flights. Phys Lett A 239(1–2):13–16
Clarke DD, Meerschaert MM, Wheatcraft SW (2005) Fractal travel time estimates for dispersive contaminants. Ground Water 43(3):401–407
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
Day TJ (1975) Longitudinal dispersion in natural channels. Water Resourc Res 11(6):909–918
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
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
Deng Z-Q, Singh VP, Bengtsson L (2004) Numerical solution of fractional advection–dispersion equation. J Hydraulic Eng 130(5):422–431
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
Di Toro DM (2001) Sediment flux modeling. J Wiley, New York, USA pp. 3–55
Falconer RA, Liu S (1988) Modeling solute transport using QUICK scheme. J Environ Eng 114(1): 3–20
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
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
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
Holly FM, Usseglio-Polatera J-D (1984) Accurate two-dimensional simulation of advective-diffusive-reactive transport. J Hydraulic Eng 127(9):728–737
Hunt B (1999) Dispersion model for mountain streams. J Hydraulic Eng 125(2):99–105
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
Komatsu T, Ohgushi K, Asai K (1997) Refined numerical scheme for advective transport in diffusion simulation. J Hydraulic Eng 123(1):41–50
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
Lin B, Falconer RA (1997) Tidal flow and transport modeling using ULTIMATE QUICKEST scheme. J Hydraulic Eng 123(4):303–314
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
Meerschaert MM, Mortensen J, Wheatcraft SW (2006) Fractional vector calculus for fractional dispersion. Phys A: Stat Mech Appl 367:181–190
Meerschaert MM, Tadjeran C (2003) Finite difference approximations for fractional dispersion flow equations. J Comput Appl Math 172:65–77
Miller KS, Ross B (1993) An introduction to the fractional calculus and fractional differential. J Wiley, New York, pp. 21–185
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
Oldham KB, Spanier J (1974) The fractional calculus. Academic Press, New York, pp. 45–134
Podlubny I (1999) Fractional differential equations. Academic Press, San Diego, pp. 41–242
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
Schumer R, Benson DA, Meerschaert MM, Baeumer B (2003) Fractal mobile/immobile solute transport. Water Resourc Res 39(10):1296:doi:10.1029/2003WR002141
Singh VP (1998) Entropy-based parameter estimation in hydrology. Kluwer Academic Publishers, Netherlands, pp. 12–39
Stefanovic DL, Stefan HD (2001) Dispersion simulation in two-dimensional tidal flow. J Hydraulic Eng 110(7):905–926
Sullivan PJ (1971) Longitudinal dispersion within a two-dimensional turbulent shear flow. J Fluid Mech 49:551–576
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
Wörman A (1998) Analytical solution and timescale for transport of reacting solutes in rivers and streams. Water Resourc Res 34(10):2703–2716
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
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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
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DOI: https://doi.org/10.1007/s10652-006-9004-5