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Fractional Dispersion, Lévy Motion, and the MADE Tracer Tests

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Abstract

The macrodispersion experiments (MADE) at the Columbus Air Force Base in Mississippi 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 contamination 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.

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References

  • Aban, I. and Meerschaert, M.: 1999, Shifted Hill's estimator for heavy tails, Preprint.

  • 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.

    Google Scholar 

  • Anderson, P. and Meerschaert, M. M.: 1998, Modeling river flows with heavy tails, Water Resour. Res. 34(9), 2271-2280.

    Google Scholar 

  • Benson, D. A.: 1998, The fractional advection-dispersion equation: Development and application, Unpublished Ph.D. thesis, Univ. of Nevada, Reno, http://www.hydro.unr.edu/homepages/ benson.

  • Benson, D. A., Wheatcraft, S. W. and Meerschaert, M. M.: 1999a, The fractional-order governing equation of Lévy motion, Preprint, http://www.hydro.unr.edu/homepages/benson.

  • Benson, D. A., Wheatcraft, S. W. and Meerschaert, M. M.: 1999b, Application of a fractional advection-dispersion equation, Preprint, http://www.hydro.unr.edu/homepages/benson.

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

    Google Scholar 

  • Berkowitz, B. and Scher, H.: 1998, Theory of anomalous chemical transport in random fracture networks, Phys. Rev. E 57(5), 5858-5869.

    Google Scholar 

  • 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.

  • 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.

    Google Scholar 

  • 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 

  • 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.

    Google Scholar 

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

    Google Scholar 

  • Compte, A. and Caceres, M. O.: 1998, Fractional dynamics in random velocity fields, Phys. Rev. Lett. 81, 3140-3143.

    Google Scholar 

  • Crank, J.: 1975, The Mathematics of Diffusion, Oxford University Press, Oxford, Great Britain.

    Google Scholar 

  • Dagan, G.: 1984, Solute transport in heterogeneous porous formations, J. Fluid Mech. 145, 151-177.

    Google Scholar 

  • Davis, R. and Resnick, S.: 1985, Limit theory for moving averages of random variables with regularly varying tail probabilities, Ann. Probab. 13, 179-195.

    Google Scholar 

  • Debnath, L.: 1995, Integral Transforms and Their Applications, CRC Press, New York.

    Google Scholar 

  • 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.

    Google Scholar 

  • Einstein, A.: 1908, Investigations on the Theory of the Brownian Movement, translation by Dover Publications in 1956 of the original manuscript.

  • Feller, W.: 1971, An Introduction to Probability Theory and Its Applications, Volume II, 2nd ed., Wiley, New York.

    Google Scholar 

  • Fofack, H. and Nolan, J.: 1998, Tail behavior, modes and other characteristics of stable distributions, Preprint, http://www.cas.american.edu/∼jpnolan/.

  • Fogedby, H. C.: 1994, Lévy flights in random environments, Phys. Rev. Lett. 73(19), 2517-2520.

    Google Scholar 

  • 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.

    Google Scholar 

  • Fürth, R.: 1956, Notes in: Einstein, A. E., Investigations on the Theory of the Brownian Movement, translation by Dover Publications.

  • Gelhar, L. W. and Axness, C. L.: 1983, Three-dimensional stochastic analysis of macrodispersion in aquifers, Water Resour. Res. 19(1), 161-180.

    Google Scholar 

  • Gnedenko, B. V. and Kolmogorov, A. N.: 1954, Limit Distributions for Sums of Random Variables, Addison-Wesley, Reading, Mass.

    Google Scholar 

  • Gorenflo, R. and Mainardi, F.: 1998, Fractional calculus and stable probability distributions, Arch. Mech 50(3), 377-388.

    Google Scholar 

  • Grigolini, P., Rocco, A. and West, B. J.: 1999, Fractional calculus as a macroscopic manifestation of randomness, Phys. Rev. E 59, 2603.

    Google Scholar 

  • 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 

  • Hill, B.: 1975, A simple general approach to inference about the tail of a distribution, Ann. Statist. 1163-1173.

  • Hosking, J. and Wallis, J.: 1987, Parameter and quantile estimation for the generalized Pareto distribution, Technometrics 29, 339-349.

    Google Scholar 

  • 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.

    Google Scholar 

  • Janicki, A. and Weron, A.: 1994, Can one see α-stable variables and processes?, Stat. Sci. 9(1), 109-126.

    Google Scholar 

  • Klafter, J., Blumen, A. and Shlesinger, M. F.: 1987, Stochastic pathway to anomalous diffusion, Phys. Rev. A 35(7), 3081-3085.

    Google Scholar 

  • LeBlanc, D. R., Garabedian, S. P., 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.

    Google Scholar 

  • Lévy, P.: 1937, Théorie de L'addition des Variables Aléatoires, Gauthier-Villars, Paris.

    Google Scholar 

  • 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.

  • Liu, H. H. and Molz, F. J.: 1997b, Multifractal analyses of hydraulic conductivity distributions, Water Resour. Res. 33(11) 2483-2488.

    Google Scholar 

  • Mandelbrot, B.: 1963, The variation of certain speculative prices, J. Business 36, 394-419.

    Google Scholar 

  • Mantegna, R. N. and Stanley, H. E.: 1995, Ultra-slow convergence to a Gaussian: the truncated Lévy flight, in: M. F. Shlesinger, G. M. Zaslavsky and U. Frisch (eds), Lévy Flights and Related Topics in Physics, Springer-Verlag, pp. 301-312.

  • McCulloch, J. H.: 1986, Simple consistent estimators of stable distribution parameters, Comm. Statist. Simul. Comput. 15, 1109-1136.

    Google Scholar 

  • McCulloch, J. H.: 1997, Measuring tail thickness to estimate the stable index alpha: A critique, J. Business Econ. Statist. 15, 74-81.

    Google Scholar 

  • Meerschaert, M.: 1986, Regular variation and domains of attraction in ℝk, Stat. Prob. Lett. 4, 43-45.

    Google Scholar 

  • 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.

    Google Scholar 

  • Meerschaert, M. M., Benson, D. A. and Bäumer, B.: 1999, Multidimensional advection and fractional dispersion, Phys. Rev. E 59(5) 5026-5028.

    Google Scholar 

  • 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.

    Google Scholar 

  • Metzler, R., Barkai, E. and Klafter, J.: 1999, Deriving fractional Fokker-Planck equations from a generalized master equation, Europhys. Lett. 46, 431-436.

    Google Scholar 

  • 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 

  • Nolan, J.: 1997, Numerical calculation of stable densities and distribution functions: Heavy tails and highly volatile phenomena, Comm. Statist. Stoch. Models 13, 759-774.

    Google Scholar 

  • Nolan, J.: 1998, Parameterizations and modes of stable distributions, Statist. Probab. Lett. 38(2), 187-195.

    Google Scholar 

  • Oldham, K. B. and Spanier, J.: 1974, The Fractional Calculus, Academic Press, New York.

    Google Scholar 

  • Pachepsky, Y. A.: 1998, Transport of water and chemicals in soils as in fractal media, Agronomy Abstracts, p. 202.

  • Pachepsky, Y. A., Benson, D. A. and Rawls, W.: 1999, Simulating scale-dependent solute transport in soils with the fractional advective-dispersive equation, Preprint.

  • Painter, S.: 1996a, Evidence for non-Gaussian scaling behavior in heterogeneous sedimentary formations, Water Resour. Res. 32(5), 1183-1195.

    Google Scholar 

  • Painter, S.: 1996b, Stochastic interpolation of aquifer properties using fractional Lévy motion, Water Resour. Res. 32(5), 1323-1332.

    Google Scholar 

  • Painter, S.: 1997, Reply to comment on 'Evidence for non-Gaussian scaling behavior in heterogeneous sedimentary formations' by H. H. Liu and F. J. Molz, Water Resour. Res. 33(4) 909-910.

  • Rajaram, H. and Gelhar, L. W.: 1991, Three-dimensional spatial moments analysis of the Borden tracer test, Water Resour. Res. 27(6), 1239-1251.

    Google Scholar 

  • 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), 3309-3324.

    Google Scholar 

  • Rocco, A. and West, B. J.: 1999, Physica A 265, 535.

    Google Scholar 

  • Ross, S.: 1988, A First Course in Probability, 5th ed., Prentice Hall, NY.

    Google Scholar 

  • Saichev, A. I. and Zaslavsky, G. M.: 1997, Fractional kinetic equations: solutions and applications, Chaos 7(4), 753-764.

    Google Scholar 

  • Samko, S. G., Kilbas, A. A. and Marichev, O. I.: 1993, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach.

  • Samorodnitsky, G. and Taqqu, M. S.: 1994, Stable Non-Gaussian Random Processes: Stochastic Models with Infinite Variance, Chapman and Hall, New York.

    Google Scholar 

  • Schumer, R., Benson, D., Meerschaert, M. and Wheatcraft, S.: 1999, A physical derivation of the fractional advection-dispersion equation, Preprint, http://www.hydro.unr.edu/homepages/ benson.

  • Serrano, S. E.: 1995, Forecasting scale-dependent dispersion from spills in heterogeneous aquifers, J. Hyd. 169, 151-169.

    Google Scholar 

  • Sheshadri, V. and West, B. J.: 1982, Fractal dimensionality of Levy processes, Proc. Natl. Acad. Sci. 79, 4501-4505.

    Google Scholar 

  • 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 

  • 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.

    Google Scholar 

  • Taylor, Sir, G. I.: 1953, Dispersion of soluble matter in solvent flowing slowly through a tube, Proc. R. Soc. A., London 219, 186-203.

    Google Scholar 

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

    Google Scholar 

  • Zheng, C. and Jiao, J. J.: 1998, Numerical simulation of tracer tests in heterogeneous aquifer, J. Environ. Eng. 124(6), 510-516.

    Google Scholar 

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Authors and Affiliations

  1. Division of Hydrologic Science, Desert Research Institute, 2215 Raggio Parkway, Reno, NV, 89512, U.S.A

    David A. Benson & Rina Schumer

  2. Department of Mathematics, University of Nevada, Reno, NV, 89557, U.S.A

    Mark M. Meerschaert

  3. Department of Geologic Sciences, University of Nevada, Reno, NV, 89557, U.S.A

    Stephen W. Wheatcraft

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  1. David A. Benson
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  4. Stephen W. Wheatcraft
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Benson, D.A., Schumer, R., Meerschaert, M.M. et al. Fractional Dispersion, Lévy Motion, and the MADE Tracer Tests. Transport in Porous Media 42, 211–240 (2001). https://doi.org/10.1023/A:1006733002131

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