Fractional Dispersion, Lévy Motion, and the MADE Tracer Tests



The macrodispersion experiments (MADE) at the Columbus Air Force Base in Missis-sippi were conducted in a highly heterogeneous aquifer that violates the basic assumptions of local second-order theories. A governing equation that describes particles that undergo Lévy motion, rather than Brownian motion, readily describes the highly skewed and heavy-tailed plume development at the MADE site. The new governing equation is based on a fractional, rather than integer, order of differentiation. This order (α), based on MADE plume measurements, is approximately 1.1. The hydraulic conductivity (K) increments also follow a power law of order α = 1.1. We conjecture that the heavy-tailed K distribution gives rise to a heavy-tailed velocity field that directly implies the fractional-order governing equation derived herein. Simple arguments lead to accurate estimates of the velocity and dispersion constants based only on the aquifer hydraulic properties. This supports the idea that the correct governing equation can be accurately determined before, or after, a contamin-ation event. While the traditional ADE fails to model a conservative tracer in the MADE aquifer, the fractional equation predicts tritium concentration profiles with remarkable accuracy over all spatial and temporal scales.

Key words

fractional derivative fractional Laplacian anomalous dispersion Lévy motion α-stable heavy tails Fokker—Planck equation MADE site. 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Aban, I. and Meerschaert, M.: 1999, Shifted Hill’s estimator for heavy tails, Preprint.Google Scholar
  2. Adams, E. E. and Gelhar, L. W.: 1992, Field study of dispersion in a heterogeneous aquifer, 2, Spatial moments analysis, Water Resour. Res. 28 (12), 3293 - 3307.CrossRefGoogle Scholar
  3. Anderson, P. and Meerschaert, M. M.: 1998, Modeling river flows with heavy tails, Water Resour. Res. 34 (9), 2271 - 2280.CrossRefGoogle Scholar
  4. Benson, D. A.: 1998, The fractional advection-dispersion equation: Development and application, Unpublished Ph.D. thesis, Univ. of Nevada, Reno, benson.Google Scholar
  5. Benson, D. A., Wheatcraft, S. W. and Meerschaert, M. M.: 1999a, The fractional-order governing equation of Lévy motion, Preprint, Scholar
  6. Benson, D. A., Wheatcraft, S. W. and Meerschaert, M. M.: 1999b, Application of a fractional advection-dispersion equation, Preprint, Scholar
  7. Berkowitz, B. and Scher, H.: 1995, On characterization of anomalous dispersion in porous and fractured media, Water Resour. Res. 31 (6), 1461 - 1466.CrossRefGoogle Scholar
  8. Berkowitz, B. and Scher, H.: 1998, Theory of anomalous chemical transport in random fracture networks, Phys. Rev. E 57 (5), 5858 - 5869.CrossRefGoogle Scholar
  9. Bhattacharya, R. and Gupta, V. K.: 1990, Application of central limit theorems to solute transport in saturated porous media: from kinetic to field scales, Chapter IV, in: J. H. Cushman (ed.), Dynamics of Fluids in Hierarchical Porous Media, Academic Press.Google Scholar
  10. Boggs, J. M. and Adams, E. E.: 1992, Field study of dispersion in a heterogeneous aquifer, 4; Investigation of adsorption and sampling bias, Water Resour. Res. 28 (12), 3325 - 3336.CrossRefGoogle Scholar
  11. Boggs, J. M., Beard, L. M., Long, S. E. and McGee, M. P.: 1993, Database for the second macrodispersion experiment (MADE-2), EPRI report TR-102072, Electric Power Res. Inst., Palo Alto, CA.Google Scholar
  12. Brusseau, M.: 1992, Transport of rate-limited sorbing solutes in heterogeneous porous media: Application of a one-dimensional multifactor nonideality model to field data, Water Resour. Res. 28 (9), 2485 - 2497.CrossRefGoogle Scholar
  13. Compte, A.: 1996, Stochastic foundations of fractional dynamics, Phys. Rev. E 53(4), 4191-4193. Compte, A. and Caceres, M. 0.: 1998, Fractional dynamics in random velocity fields, Phys. Rev. Lett. 81, 3140 - 3143.CrossRefGoogle Scholar
  14. Crank, J.: 1975, The Mathematics of Diffusion, Oxford University Press, Oxford, Great Britain. Dagan, G.: 1984, Solute transport in heterogeneous porous formations, J. Fluid Mech. 145, 151 - 177.Google Scholar
  15. Davis, R. and Resnick, S.: 1985, Limit theory for moving averages of random variables with regularly varying tail probabilities, Ann. Probab. 13, 179 - 195.CrossRefGoogle Scholar
  16. Debnath, L.: 1995, Integral Transforms and Their Applications, CRC Press, New York.Google Scholar
  17. Deng, F.-W., Cushman, J. H. and Delleur, J. W.: 1993, A fast Fourier transform stochastic analysis of the contaminant transport problem, Water Resour. Res. 29 (9), 3241 - 3247.CrossRefGoogle Scholar
  18. Einstein, A.: 1908, Investigations on the Theory of the Brownian Movement,translation by Dover Publications in 1956 of the original manuscript.Google Scholar
  19. Feller, W.: 1971, An Introduction to Probability Theory and Its Applications, Volume II, 2nd ed., Wiley, New York.Google Scholar
  20. Fofack, H. and Nolan, J.: 1998, Tail behavior, modes and other characteristics of stable distributions, Preprint, Google Scholar
  21. Fogedby, H. C.: 1994, Lévy flights in random environments, Phys. Rev. Lett. 73 (19), 2517 - 2520.CrossRefGoogle Scholar
  22. Freyberg, D. L.: 1986, A natural gradient experiment on solute transport in a sandy aquifer, 2, Spatial moments and the advection and dispersion of nonreactive tracers, Water Resour. Res. 22 (13), 2031 - 2046.CrossRefGoogle Scholar
  23. Fürth, R.: 1956, Notes in: Einstein, A. E., Investigations on the Theory of the Brownian Movement, translation by Dover Publications.Google Scholar
  24. Gelhar, L. W. and Axness, C. L.: 1983, Three-dimensional stochastic analysis of macrodispersion in aquifers, Water Resour. Res. 19 (1), 161 - 180.CrossRefGoogle Scholar
  25. Gnedenko, B. V. and Kolmogorov, A. N.: 1954, Limit Distributions for Sums of Random Variables, Addison-Wesley, Reading, Mass.Google Scholar
  26. Gorenflo, R. and Mainardi, F.: 1998, Fractional calculus and stable probability distributions, Arch. Mech 50 (3), 377 - 388.Google Scholar
  27. Grigolini, P., Rocco, A. and West, B. J.: 1999, Fractional calculus as a macroscopic manifestation of randomness, Phys. Rev. E 59, 2603.CrossRefGoogle Scholar
  28. Haggerty, R. and Gorelick, S. M.: 1995, Multiple-rate mass transfer for modeling diffusion and surface reactions in media with pore-scale heterogeneity, Water Resour. Res. 31 (10), 2383 - 2400.Google Scholar
  29. Hill, B.: 1975, A simple general approach to inference about the tail of a distribution, Ann. Statist. 1163 - 1173.Google Scholar
  30. Hosking, J. and Wallis, J.: 1987, Parameter and quantile estimation for the generalized Pareto distribution, Technometrics 29, 339 - 349.CrossRefGoogle Scholar
  31. Hughes, B. D., Shlesinger, M. F. and Montroll, E. W.: 1981, Random walks with self-similar clusters, Proc. Natl. Acad. Sci. USA 78 (6), 3287 - 3291.CrossRefGoogle Scholar
  32. Janicki, A. and Weron, A.: 1994, Can one see a-stable variables and processes?, Stat. Sci. 9 (1), 109 - 126.CrossRefGoogle Scholar
  33. Klafter, J., Blumen, A. and Shlesinger, M. F.: 1987, Stochastic pathway to anomalous diffusion, Phys. Rev. A 35 (7), 3081 - 3085.Google Scholar
  34. LeBlanc, D. R., Garabedian, S. R, Hess, K. M., Gelhar, L. W., Quadri, R. D., Stollenwerk, K. G. and Wood, W. W.: 1991, Large-scale natural gradient tracer test in sand and gravel, Cape Cod, Massachusetts, 1, Experimental design and observed tracer movement, Water Resour. Res. 27 (5), 895 - 910.CrossRefGoogle Scholar
  35. Lévy, R: 1937, Théorie de L’addition des Variables Aléatoires, Gauthier-Villars, Paris.Google Scholar
  36. Liu, H. H. and Molz, F. J.: 1997a, Comment on ‘Evidence for non-Gaussian scaling behavior in heterogeneous sedimentary formations’ by Scott Painter, Water Resour. Res. 33 (4) 907 - 908.CrossRefGoogle Scholar
  37. Liu, H. H. and Molz, F. J.: 1997b, Multifractal analyses of hydraulic conductivity distributions, Water Resour. Res. 33 (11) 2483 - 2488.CrossRefGoogle Scholar
  38. Mandelbrot, B.: 1963, The variation of certain speculative prices, J. Business 36, 394 - 419.CrossRefGoogle Scholar
  39. Mantegna, R. N. and Stanley, H. E.: 1995, Ultra-slow convergence to a Gaussian: the truncated Lévy flight, in: M. F. Shlesinger, G. M. Zaslaysky and U. Frisch (eds), Lévy Flights and Related Topics in Physics, Springer-Verlag, pp. 301 - 312.Google Scholar
  40. McCulloch, J. H.: 1986, Simple consistent estimators of stable distribution parameters, Comm. Statist. Simul. Comput. 15, 1109 - 1136.CrossRefGoogle Scholar
  41. McCulloch, J. H.: 1997, Measuring tail thickness to estimate the stable index alpha: A critique, J. Business Econ. Statist. 15, 74 - 81.Google Scholar
  42. Meerschaert, M.: 1986, Regular variation and domains of attraction in R1`, Stat. Prob. Lett. 4, 43 - 45.CrossRefGoogle Scholar
  43. Meerschaert, M. and Scheffler, H.-P.: 1998, A simple robust estimator for the thickness of heavy tails, J. Stat. Plann. Inference 71 (1-2), 19 - 34.CrossRefGoogle Scholar
  44. Meerschaert, M. M., Benson, D. A. and Bäumer, B.: 1999, Multidimensional advection and fractional dispersion, Phys. Rev. E 59 (5) 5026 - 5028.CrossRefGoogle Scholar
  45. Metzler, R., Klafter, J. and Sokolov, I. M.: 1998, Anomalous transport in external fields: Continuous time ransom walks and fractional diffusion equations extended, Phys. Rev. E 58, 1621 - 1633.CrossRefGoogle Scholar
  46. Metzler, R., Barkai, E. and Klafter, J.: 1999, Deriving fractional Fokker-Planck equations from a generalized master equation, Europhys. Lett. 46, 431 - 436.CrossRefGoogle Scholar
  47. Molz, F. J., Liu, H. H. and Szulga, J.: 1997, Fractional Brownian motion and fractional Gaussian noise in subsurface hydrology: A review, presentation of fundamental properties, and extensions, Water Resour. Res. 33(10), 2273-2286.Google Scholar
  48. Nolan, J.: 1997, Numerical calculation of stable densities and distribution functions: Heavy tails and highly volatile phenomena, Comm. Statist. Stock. Models 13, 759 - 774.CrossRefGoogle Scholar
  49. Nolan, J.: 1998, Parameterizations and modes of stable distributions, Statist. Probab. Lett. 38 (2), 187 - 195.CrossRefGoogle Scholar
  50. Oldham, K. B. and Spanier, J.: 1974, The Fractional Calculus,Academic Press, New York. Pachepsky, Y. A.: 1998, Transport of water and chemicals in soils as in fractal media, Agronomy Abstracts,p. 202.Google Scholar
  51. Pachepsky, Y. A., Benson, D. A. and Rawls, W.: 1999, Simulating scale-dependent solute transport in soils with the fractional advective-dispersive equation, Preprint.Google Scholar
  52. Painter, S.: 1996a, Evidence for non-Gaussian scaling behavior in heterogeneous sedimentary formations, Water Resour. Res. 32 (5), 1183 - 1195.CrossRefGoogle Scholar
  53. Painter, S.: 1996b, Stochastic interpolation of aquifer properties using fractional Lévy motion, Water Resour. Res. 32 (5), 1323 - 1332.CrossRefGoogle Scholar
  54. Painter, S.: 1997, Reply to comment on ‘Evidence for non-Gaussian scaling behavior in heterogeneous sedimentary formations’ by H. H. Liu and E J. Molz, Water Resour. Res. 33 (4) 909 - 910.CrossRefGoogle Scholar
  55. Rajaram, H. and Gelhar, L. W.: 1991, Three-dimensional spatial moments analysis of the Borden tracer test, Water Resour. Res. 27 (6), 1239 - 1251.CrossRefGoogle Scholar
  56. Rehfeldt, K. R., Boggs, J. M. and Gelhar, L. W.: 1992, Field study of dispersion in a heterogeneous aquifer. 3: Geostatistical analysis of hydraulic conductivity, Water Resour. Res. 28(12), 33093324.Google Scholar
  57. Rocco, A. and West, B. J.: 1999, Physica A 265, 535.CrossRefGoogle Scholar
  58. Ross, S.: 1988, A First Course in Probability, 5th ed., Prentice Hall, NY.Google Scholar
  59. Saichev, A. I. and Zaslaysky, G. M.: 1997, Fractional kinetic equations: solutions and applications, Chaos 7 (4), 753 - 764.CrossRefGoogle Scholar
  60. Samko, S. G., Kilbas, A. A. and Marichev, O. I.: 1993, Fractional Integrals and Derivatives: Theory and Applications,Gordon and Breach.Google Scholar
  61. Samorodnitsky, G. and Taqqu, M. S.: 1994, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance, Chapman and Hall, New York.Google Scholar
  62. Schumer, R., Benson, D., Meerschaert, M. and Wheatcraft, S.: 1999, A physical derivation of the fractional advection-dispersion equation, Preprint, benson.Google Scholar
  63. Serrano, S. E.: 1995, Forecasting scale-dependent dispersion from spills in heterogeneous aquifers, J. Hyd. 169, 151 - 169.CrossRefGoogle Scholar
  64. Sheshadri, V. and West, B. J.: 1982, Fractal dimensionality of Levy processes, Proc. Natl. Acad. Sci. 79, 4501 - 4505.CrossRefGoogle Scholar
  65. Shlesinger, M. F., Klafter, J. and Wong, Y. M.: 1982, Random walks with infinite spatial and temporal moments, J. Stat. Phys. 27(3), 499-512.Google Scholar
  66. Sudicky, E. A.: 1986, A natural gradient experiment on solute transport in a sandy aquifer: Spatial variability of hydraulic conductivity and its role in the dispersion process, Water Resour. Res. 22 (13), 2069 - 2082.CrossRefGoogle Scholar
  67. Taylor, Sir, G. I.: 1953, Dispersion of soluble matter in solvent flowing slowly through a tube, Proc. R. Soc. A., London 219, 186 - 203.CrossRefGoogle Scholar
  68. Zaslaysky, G. M.: 1994, Renormalization group theory of anomalous transport in systems with Hamiltonian chaos, Chaos 4 (1), 25 - 33.CrossRefGoogle Scholar
  69. Zheng, C. and Jiao, J. J.: 1998, Numerical simulation of tracer tests in heterogeneous aquifer, J. Environ. Eng. 124 (6), 510 - 516.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  1. 1.Division of Hydrologic ScienceDesert Research InstituteRenoUSA
  2. 2.Department of MathematicsUniversity of NevadaRenoUSA
  3. 3.Department of Geologic SciencesUniversity of NevadaRenoUSA

Personalised recommendations