A mass balance based numerical method for the fractional advection-dispersion equation: Theory and application

Water Resources Research

Volumn 41, Issue 7, 2005, Pages 1-10

  Zhang, Xiaoxian ^a   Crawford, John W ^a   Deeks, Lynda K ^b   Stutter, Marc I ^c   Bengough, A Glyn ^c   Young, Iain M ^a  

^a UNIVERSITY OF ABERTAY DUNDEE   (United Kingdom)

^b CRANFIELD UNIVERSITY   (United Kingdom)

^c SCOTTISH CROP RESEARCH INSTITUTE   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

BOUNDARY CONDITIONS; HYDROLOGY; LINEAR SYSTEMS; NUMERICAL METHODS; PROBLEM SOLVING; WATER; WATER RESOURCES;

CHEMICAL MOVEMENTS; CONTINUOUS TIME RANDOM WALKS (CTRW); SEMI-IMPLICIT SCHEMES; SLOW PATHWAYS;

GEOPHYSICS;

ADVECTION-DIFFUSION EQUATION; HETEROGENEITY; MASS BALANCE; SOLUTE TRANSPORT;

EID: 23944449029     PISSN: 00431397     EISSN: None     Source Type: Journal    
DOI: 10.1029/2004WR003818     Document Type: Article

References (45)

1. Bear, J. (1979), Hydraulics of Groundwater, McGraw-Hill, New York.
2. Benson, D. A. (2001), A model of water streaking down a wall, Water Resour. Res., 37(2), 427-430.
3. Benson, D. A., S. W. Wheatcraft, and M. M. Meerschaert (2000a), Application of a fractional advection-dispersion equation, Water Resour. Res., 36(6), 1403-1412.
4. Benson, D. A., S. W. Wheatcraft, and M. M. Meerschaert (2000b), The fractional-order governing equation of Levy motion, Water Resour. Res., 36(6), 1413-1423.
5. Berkowitz, B., and H. Scher (1997), Anomalous transport in random fracture networks, Phys. Rev. Lett., 79(20), 4038-4041.
6. Berkowitz, B., and H. Scher (1998), Theory of anomalous chemical transport in random fracture networks, Phys. Rev. E, 57(5), 5858-5869.
7. Berkowitz, B., H. Scher, and S. E. Silliman (2000), Anomalous transport in laboratory-scale, heterogeneous porous media, Water Resour. Res., 36(1), 149-158.
8. Berkowitz, B., J. Klafter, R. Metzler, and H. Scher (2002), Physical pictures of transport in heterogeneous media: Advection-dispersion, random-walk, and fractional derivative formulations, Water Resour. Res., 38(10), 1191, doi: 10.1029/2001WR001030.
9. Bromly, M., and C. Hinz (2004), Non-Fickian transport in homogeneous unsaturated repacked sand, Water Resour. Res., 40, W07402, doi: 10.1029/2003WR002579.
10. Caputo, M. (1967), Linear models of dissipation whose Q is almost frequency independent-II, Geophys. J. R. Astron. Soc., 13, 529-539.
11. Caputo, M., and F. Mainardi (1971), A new dissipation model based on memory mechanism, Pure Appl. Geophys., 91, 134-147.
12. Coats, K. H., and B. D. Smith (1964), Dead-end pore volume and dispersion in porous media, Soc. Pet. Eng. J., 4(1), 73-84.
13. Cortis, A., and B. Berkowitz (2004), Anomalous transport in "classical" soil and sand columns, Soil Sci. Soc. Am. J., 68(5), 1539-1548.
14. Cortis, A., C. Gallo, H. Scher, and B. Berkowitz (2004a), Numerical simulation of non-Fickian transport in geological formations with multiple-scale heterogeneities, Water Resour. Res., 40, W04209, doi: 10.1029/2003WR002750.
15. Cortis, A., Y. J. Chen, H. Scher, and B. Berkowitz (2004b), Quantitative characterization of pore-scale disorder effects on transport in "homogeneous" granular media, Phys. Rev. E, 70(4), doi: 10.1103/PhysRevE.70.041108.
16. Cushman, J. H., and T. R. Ginn (2000), Fractional advection-dispersion equation: A classical mass balance with convolution-Fickian flux, Water Resour. Res., 36(12), 3763-3766.
17. Cushman, J. H., B. X. Hu, and T. R. Ginn (1994), Nonequilibrium statistical mechanics of preasymptotic dispersion, J. Stat. Phys., 75, 859-878.
18. Decks, L. K., A. G. Bengough, D. Low, M. F. Billett, X. Zhang, J. W. Crawford, J. M. Chessell, and I. M. Young (2004), Spatial variation of effective porosity and its implications for discharge in an upland head-water catchment in Scotland, J. Hydrol., 290(3-4), 217-228.
19. Deng, Z. Q., V. P. Singh, and L. Bengtsson (2004), Numerical solution of fractional advection-dispersion equation, J. Hydraul. Eng., 130(5), 422-431.
20. Dentz, M., A. Cortis, H. Scher, and B. Berkowitz (2004), Time behavior of solute transport in heterogeneous media: Transition from anomalous to normal transport, Adv. Water Resour., 27(2), 155-173.
21. Ellsworth, T. R., P. J. Shouse, T. H. Skaggs, J. A. Jobes, and J. Fargerlund (1996), Solute transport in unsaturated soil: Experimental design, parameter estimation, and model discrimination, Soil Sci. Soc. Am. J., 60(2), 397-407.
22. Fischer, H. B., E. J. List, R. C. Y. Koh, J. Imberger, and N. H. Brooks (1979), Mixing in Inland and Coastal Waters, Elsevier, New York.
23. Haggerty, R., C. F. Harvey, C. Freiherr von Schwerin, and L. C. Meigs (2004), What controls the apparent timescale of solute mass transfer in aquifers and soils? A comparison of experimental results, Water Resour. Res., 40, W01510, doi: 10.1029/2002WR001716.
24. Hunt, B. (1999), Dispersion model for mountain streams, J. Hydraul. Eng., 125(2), 99-105.
25. Jobson, H. E. (2001), Predicting river travel time from hydraulic characteristics, J. Hydraul. Eng., 127(11), 911-918.
26. Kirchner, J. W., X. H. Feng, and C. Neal (2000), Fractal stream chemistry and its implications for contaminant transport in catchments, Nature, 403(6169), 524-527.
27. Kosakowski, G., B. Berkowitz, and H. Scher (2001), Analysis of field observations of tracer transport in a fractured till, J. Contam. Hydrol., 47(1), 29-51.
28. Liu, F., V. Anh, and I. Turner (2004), Numerical solution of the space fractional Fokker-Planck equation, J. Comput. Appl. Math., 166(1), 209-219.
29. Lynch, V. E., B. A. Carreras, D. del-Castillo-Negrete, K. M. Ferreira-Mejias, and H. R. Hicks (2003), Numerical methods for the solution of partial differential equations of fractional order, J. Comput. Phys., 192, 406-421.
30. Meerschaert, M. M., and C. Tadjeran (2004), Finite difference approximations for fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172(1), 65-77.
31. Meerschaert, M. M., D. A. Benson, and B. Baumer (1999), Multidimensional advection and fractional dispersion, Phys. Rev. E, 59(5), 5026-5028.
32. Neuman, S. P. (1984), Adaptive Eulerian-Lagrangian finite element method for advection-dispersion, Int. J. Numer. Methods Eng., 20(2), 321-337.
33. Nolan, J. (2005), Stable Distributions: Models for Heavy Tailed Data, Springer, in press.
34. Nordin, C. F., and B. M. Troutman (1980), Longitudinal dispersion in rivers: The persistence of skewness in observed data, Water Resour. Res., 16(1), 123-128.
35. Pachepsky, Y., D. Benson, and W. Rawls (2000), Simulating scale-dependent solute transport in soils with the fractional advective-dispersive equation, Soil Sci. Soc. Am. J., 64(4), 1234-1243.
36. Pang, L. P., and B. Hunt (2001), Solutions and verification of a scale-dependent dispersion model, J. Contam. Hydrol., 53(1-2), 21-39.
37. Samko, S. G., A. A. Kilbas, and O. I. Marichev (1993), Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York.
38. Scher, H., G. Margolin, R. Metzler, J. Klafter, and B. Berkowitz (2002), The dynamical foundation of fractal stream chemistry: The origin of extremely long retention times, Geophys. Res. Lett., 29(5), 1061, doi: 10.1029/2001GL014123.
39. Schumer, R., D. A. Benson, M. M. Meerschaert, and S. W. Wheatcraft (2001). Eulerian derivation of the fractional advection-dispersion equation, J. Contam. Hydrol., 48(1-2), 69-88.
40. Schumer, R., D. A. Benson, M. M. Meerschaert, and B. Baeumer (2003), Fractal mobile/immobile solute transport, Water Resour. Res., 39(10), 1296, doi: 10.1029/2003WR002141.
41. Seredynska, M., and A. Hanyga (2000), Nonlinear Hamiltonian equations with fractional damping, J. Math. Phys., 41, 2135-2156.
42. Stutter, M. I., L. K. Deeks, and M. F. Billett (2004), Spatial variability in soil ion exchange chemistry in a granitic upland catchment, Soil Sci. Soc. Am. J., 68(4), 1304-1314.
43. Stutter, M. I., L. K. Deeks, and M. F. Billett (2005), Transport of conservative and reactive tracers through a naturally structured upland podzol field lysimeter, J. Hydrol., 300(1-4), 1-19.
44. van Genuchten, M. T., and P. J. Wierenga (1976), Mass transfer studies in sorbing porous media: 1. Analytical solutions, Soil Sci. Soc. Am. J., 40(4), 473-480.
45. Zhang, R. D., K. Huang, and J. Xiang (1994), Solute movement through homogeneous and heterogeneous soil columns, Adv. Water. Resour., 17(5), 317-324.