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Diffusion in Random Velocity Fields

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Diffusion in Random Fields

Part of the book series: Geosystems Mathematics ((GSMA))

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Abstract

Classical stochastic theories for transport in subsurface are revisited and transport models are formulated as stochastic processes. The process of diffusion with space variable drift coefficients is proposed as a general frame for stochastic modeling in subsurface hydrology. Stochastic homogeneity properties, first order approximations, and the occurrence of anomalous diffusion, ergodic, and self-averaging properties are presented.

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References

  1. Aït-Sahalia, Y.: Telling from discrete data whether the underlying continuous-time model is a diffusion. J. Financ. 57, 2075–2112 (2002)

    Google Scholar 

  2. Arnold, V.I.: Ordinary Differential Equations. Springer, Berlin (1992)

    Google Scholar 

  3. Attinger, S., Dentz, M., Kinzelbach, H., Kinzelbach, W.: Temporal behavior of a solute cloud in a chemically heterogeneous porous medium. J. Fluid Mech. 386, 77–104 (1999)

    MATH  Google Scholar 

  4. Avellaneda, M., Majda, M.: Stieltjes integral representation and effective diffusivity bounds for turbulent diffusion. Phys. Rev. Lett. 62(7), 753–755 (1989)

    Google Scholar 

  5. Avellaneda, M., Majda, M.: Superdiffusion in nearly stratified flows. J. Stat. Phys. 69(3/4), 689–729 (1992)

    MathSciNet  MATH  Google Scholar 

  6. Avellaneda, M., Elliot, F. Jr., Apelian, C.: Trapping, percolation and anomalous diffusion of particles in a two-dimensional random field. J. Stat. Phys. 72(5/6), 1227–1304 (1993)

    MathSciNet  MATH  Google Scholar 

  7. Balescu, R.: Transport Processes in Plasmas. North-Holland, Amsterdam (1988)

    MATH  Google Scholar 

  8. Balescu, R., Wang, H-D., Misguich, J.H.: Langevin equation versus kinetic equation: subdiffusive behavior of charged particles in a stochastic magnetic field. Phys. Plasmas 1(12), 3826–3842 (1994)

    Google Scholar 

  9. Bear, J.: On the tensor form of dispersion in porous media. J. Geophys. Res. 66(4), 1185–1197 (1961)

    MathSciNet  Google Scholar 

  10. Bhattacharya, R.N., Gupta, V.K.: A theoretical explanation of solute dispersion in saturated porous media at the Darcy scale. Water Resour. Res. 19, 934–944 (1983)

    Google Scholar 

  11. Bouchaud, J.-P., Georges, A.: Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys. Rep. 195, 127–293 (1990)

    MathSciNet  Google Scholar 

  12. Bouchaud, J.-P., Georges, A., Koplik, J., Provata, A., Redner, S.: Superdiffusion in random velocity fields. Phys. Rev. Lett. 64, 2503–2506 (1990)

    Google Scholar 

  13. Chilès, J.P., Delfiner, P.: Geostatistics: Modeling Spatial Uncertainty. Wiley, New York (1999)

    MATH  Google Scholar 

  14. Clincy, M., Kinzelbach, H.: Stratified disordered media: exact solutions for transport parameters and their self-averaging properties. J. Phys. A. Math. Gen. 34, 7142–7152 (2001)

    MATH  Google Scholar 

  15. Colucci, P.J., Jaberi, F.A., Givi, P.: Filtered density function for large eddy simulation of turbulent reacting flows. Phys. Fluids 10(2), 499–515 (1998)

    MathSciNet  MATH  Google Scholar 

  16. Dagan, G.: Solute transport in heterogeneous porous formations. J. Fluid Mech. 145, 151–177 (1984)

    MATH  Google Scholar 

  17. Dagan, G.: Theory of solute transport by groundwater. Annu. Rev. Fluid Mech. 19, 183–215 (1987)

    MATH  Google Scholar 

  18. Dagan, G.: Time-dependent macrodispersion for solute transport in anisotropic heterogeneous aquifers. Water Resour. Res. 24, 1491–1500 (1988)

    Google Scholar 

  19. Dagan, G.: Flow and Transport in Porous Formations. Springer, Berlin (1989)

    Google Scholar 

  20. Dagan, G.: Transport in heterogeneous porous formations: spatial moments, ergodicity, and effective dispersion. Water Resour. Res. 26(6), 1281–1290 (1990)

    Google Scholar 

  21. Deng, W., Barkai, E.: Ergodic properties of fractional Brownian-Langevin motion. Phys. Rev. E 79, 011112 (2009)

    MathSciNet  Google Scholar 

  22. Deng, F.W., Cushman, J.H.: On higher-order corrections to the flow velocity covariance tensor. Water Resour. Res. 31(7), 1659–1672 (1995)

    Google Scholar 

  23. Dentz, M., de Barros, F.P.J.: Dispersion variance for transport in heterogeneous porous media. Water Resour. Res. 49, 3443–3461 (2013)

    Google Scholar 

  24. Dentz, M., Kinzelbach, H., Attinger, S., Kinzelbach, W.: Temporal behavior of a solute cloud in a heterogeneous porous medium 1. Point-like injection. Water Resour. Res. 36, 3591–3604 (2000)

    MATH  Google Scholar 

  25. Dentz, M., Kinzelbach, H., Attinger, S., Kinzelbach, W.: Temporal behavior of a solute cloud in a heterogeneous porous medium 2. Spatially extended injection. Water Resour. Res. 36, 3605–3614 (2000)

    MATH  Google Scholar 

  26. Di Federico, V., Neuman, S.P.: Scaling of random fields by means of truncated power variograms and associated spectra. Water Resour. Res. 33, 1075–1085 (1997)

    Google Scholar 

  27. Doob, J.L.: Stochastic Processes. Wiley, New York (1990)

    MATH  Google Scholar 

  28. Dybiec, B., Gudowska-Nowak, E.: Discriminating between normal and anomalous random walks. Phys. Rev. E 80, 061122 (2009)

    Google Scholar 

  29. Eberhard, J.: Approximations for transport parameters and self-averaging properties for point-like injections in heterogeneous media. J. Phys. A. Math. Gen. 37, 2549–2571 (2004)

    MathSciNet  MATH  Google Scholar 

  30. Eberhard, J., Suciu, N., Vamos, C.: On the self-averaging of dispersion for transport in quasi-periodic random media. J. Phys. A: Math. Theor. 40, 597–610 (2007)

    MathSciNet  MATH  Google Scholar 

  31. Fannjiang, A., Komorowski, T.: Diffusive and nondiffusive limits of transport in nonmixing flows. SIAM J. Appl. Math. 62, 909–923 (2002)

    MathSciNet  MATH  Google Scholar 

  32. Fiori, A.: Finite Peclet extensions of Dagan’s solutions to transport in anisotropic heterogeneous formations. Water Resour. Res. 32, 193–198 (1996)

    Google Scholar 

  33. Fiori, A.: On the influence of local dispersion in solute transport through formations with evolving scales of heterogeneity. Water Resour. Res. 37(2), 235–242 (2001)

    Google Scholar 

  34. Fiori, A., Dagan, G.: Concentration fluctuations in aquifer transport: a rigorous first-order solution and applications. J. Contam. Hydrol. 45, 139–163 (2000)

    Google Scholar 

  35. Fried, J.J.: Groundwater Pollution. Elsevier, New York (1975)

    Google Scholar 

  36. Gardiner, C.W.: Stochastic Methods. Springer, Berlin (2009)

    MATH  Google Scholar 

  37. Gelhar, L.W.: Stochastic subsurface hydrology from theory to applications. Water Resour. Res. 22(9S), 135S–145S (1986)

    Google Scholar 

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

    Google Scholar 

  39. Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series, and Products. Elsevier, Amsterdam (2007)

    MATH  Google Scholar 

  40. Honkonen, G.: Stochastic processes with stable distributions in random environments. Phys. Rev. E 53(1), 327–331 (1996)

    MathSciNet  Google Scholar 

  41. Isichenko, M.B.: Percolation, Statistical Topography, and Transport in Random Media. Rev. Mod. Phys. 64, 961 (1992)

    MathSciNet  Google Scholar 

  42. Jaekel, U., Vereecken, H.: Renormalization group analysis of macrodispersion in a directed random flow. Water Resour. Res. 33, 2287–2299 (1997)

    Google Scholar 

  43. Jeon, J.-H., Metzler, R.: Fractional Brownian motion and motion governed by the fractional Langevin equation in confined geometries. Phys. Rev. E 81, 021103 (2010)

    MathSciNet  Google Scholar 

  44. Jury, W.A., Sposito, G.: Field calibration and validation of solute transport models for the unsaturated zone. Soil Sci. Soc. Am. J. 49, 1331–1241 (1985)

    Google Scholar 

  45. Kesten, H., Papanicolaou, G.C.: A limit theorem for turbulent diffusion. Commun. Math. Phys. 65, 97–128 (1979)

    MathSciNet  MATH  Google Scholar 

  46. Kitanidis, P.K.: Prediction by the method of moments of transport in a heterogeneous formation. J. Hydrol. 102, 453–473 (1988)

    Google Scholar 

  47. Kloeden, P.E., Platen, E.: Numerical Solutions of Stochastic Differential Equations. Springer, Berlin (1999)

    MATH  Google Scholar 

  48. Kolmogorov, A.N.: Grundbegriffe der Warscheinlichkeitsrechnung. Springer, Berlin (1933)

    Google Scholar 

  49. Le Doussal, P., Machta, J.: Annealed versus quenched diffusion coefficient in random media. Phys. Rev. B 40(12), 9427–9430 (1989)

    Google Scholar 

  50. Lumley, J.L.: The mathematical nature of the problem of relating Lagrangian and Eulerian statistical functions in turbulence. In: Mécanique de la Turbulence. Coll. Intern. du CNRS à Marseille (Ed. CNRS, Paris, 1962), pp. 17–26

    Google Scholar 

  51. Majda, A.J., Kramer, P.R.: Simplified models for turbulent diffusion: theory, numerical modelling, and physical phenomena. Phys. Rep. 14, 237–574 (1999)

    MathSciNet  Google Scholar 

  52. Majumdar, S.N.: Persistence of a particle in the Matheron–de Marsily velocity field. Phys. Rev. E 68, 050101(R) (2003)

    Google Scholar 

  53. Matheron, G., de Marsily, G.: Is transport in porous media always diffusive? Water Resour. Res. 16, 901–917 (1980)

    Google Scholar 

  54. Monin, A.S., Yaglom, A.M.: Statistical Fluid Mechanics, Volume II: Mechanics of Turbulence. MIT Press, Cambridge (1975)

    Google Scholar 

  55. O’Malley, D., Cushman, J.H.: A renormalization group classification of nonstationary and/or infinite second moment diffusive processes. J. Stat. Phys. 146(5), 989–1000 (2012)

    MathSciNet  MATH  Google Scholar 

  56. O’Malley, D., Cushman, J.H.: Two scale renormalization group classification of diffusive processes. Phys. Rev. E 86(1), 011126 (2012)

    Google Scholar 

  57. Papoulis, A., Pillai, S. U.: Probability, Random Variables and Stochastic Processes. McGraw-Hill, Singapore (2009)

    Google Scholar 

  58. Pope, S.B.: PDF methods for turbulent reactive flows. Prog. Energy Combust. Sci. 11(2), 119–192 (1885)

    Google Scholar 

  59. Port, S.C., Stone, C.J.: Random measures and their application to motion in an incompressible fluid. J. Appl. Prob. 13, 498–506 (1976)

    MathSciNet  MATH  Google Scholar 

  60. Reuss, J.-D., Misguish, J.H.: Low frequency percolation scaling for particle diffusion in electrostatic turbulence. Phys. Rev. E 54(2), 1857–1869 (1996)

    Google Scholar 

  61. Ross, K., Attinger, S.: Temporal behaviour of a solute cloud in a fractal heterogeneous porous medium at different scales. Geophys. Res. Abstr. 12, EGU2010-10921-2 (2010)

    Google Scholar 

  62. Russo, D.: On the velocity covariance and transport modeling in heterogeneous anisotropic porous formations. Water Resour. Res. 31(1), 129–137 (1995)

    Google Scholar 

  63. Russo, D.: A note on ergodic transport of a passive solute in partially saturated porous formations. Water Resour. Res. 32(12), 3623–3628 (1996)

    Google Scholar 

  64. Saffman, P.G.: Application of the Wiener-Hermite expansion to the diffusion of a passive scalar in a homogeneous turbulent flow. Phys. Fluids 12(9), 1786–1798 (1969)

    MathSciNet  MATH  Google Scholar 

  65. Scheidegger, A.E.: Statistical hydrodynamics in porous media. J. Appl. Phys. 25(8), 994–1001 (1954)

    MathSciNet  MATH  Google Scholar 

  66. Scheidegger, A.E.: General theory of dispersion in porous media. J. Geophys. Res. 66(10), 3273–3278 (1961)

    Google Scholar 

  67. Schwarze, H., Jaekel, U., Vereecken, H.: Estimation of macrodispersivity by different approximation methods for flow and transport in randomly heterogeneous media. Transp. Porous Media 43, 265–287 (2001)

    Google Scholar 

  68. Shapiro, A.M., Cvetkovic, V.D.: Stochastic analysis of solute travel time in heterogeneous Porous media. Water Resour. Res. 24(10), 1711–1718 (1988)

    Google Scholar 

  69. Suciu, N.: Spatially inhomogeneous transition probabilities as memory effects for diffusion in statistically homogeneous random velocity fields. Phys. Rev. E 81, 056301 (2010)

    Google Scholar 

  70. Suciu, N.: Diffusion in random velocity fields with applications to contaminant transport in groundwater. Adv. Water Resour. 69, 114–133 (2014)

    Google Scholar 

  71. Suciu, N., Vamoş, C.: Comment on “Nonstationary flow and nonergodic transport in random porous media” by G. Darvini and P. Salandin. Water Resour. Res. 43, W12601 (2007)

    Google Scholar 

  72. Suciu, N., Vamoş, C.: Ergodic estimations of upscaled coefficients for diffusion in random velocity fields. In: L’Ecuyér, P., Owen, A.B. (eds.) Monte Carlo and Quasi-Monte Carlo Methods 2008, pp. 617–626. Springer, Berlin (2009)

    Google Scholar 

  73. Suciu, N., Vamoş, C., Vanderborght, J., Hardelauf, H., Vereecken, H.: Numerical investigations on ergodicity of solute transport in heterogeneous aquifers. Water Resour. Res. 42, W04409 (2006)

    MATH  Google Scholar 

  74. Suciu, N., Vamoş, C., Eberhard, J.: Evaluation of the first-order approximations for transport in heterogeneous media. Water Resour. Res. 42, W11504 (2006)

    Google Scholar 

  75. Suciu, N., Vamos, C., Vereecken, H., Sabelfeld, K., Knabner, P.: Memory effects induced by dependence on initial conditions and ergodicity of transport in heterogeneous media. Water Resour. Res. 44, W08501 (2008)

    MATH  Google Scholar 

  76. Suciu, N., Vamos, C., Vereecken, H., Sabelfeld, K., Knabner, P.: Ito equation model for dispersion of solutes in heterogeneous media. Rev. Anal. Num. Theor. Approx. 37, 221–238 (2008)

    MathSciNet  MATH  Google Scholar 

  77. Suciu, N., Vamos, C., Radu, F.A., Vereecken, H., Knabner, P.: Persistent memory of diffusing particles. Phys. Rev. E 80, 061134 (2009)

    Google Scholar 

  78. Suciu, N., Attinger, S., Radu, F.A., Vamoş, C., Vanderborght, J., Vereecken, H., Knabner, P.: Solute transport in aquifers with evolving scale heterogeneity. Print No. 346, Mathematics Department—Friedrich-Alexander University Erlangen-Nuremberg (2011)

    Google Scholar 

  79. Suciu, N., Attinger, S., Radu, F.A., Vamoş, C., Vanderborght, J., Vereecken, H. Knabner, P.: Solute transport in aquifers with evolving scale heterogeneity. Analele Universitatii “Ovidius” Constanta-Seria Matematica 23(3), 167–186 (2015)

    MathSciNet  MATH  Google Scholar 

  80. Suciu, N., Schüler, L., Attinger, S., Knabner, P.: Towards a filtered density function approach for reactive transport in groundwater. Adv. Water Resour. 90, 83–98 (2016)

    Google Scholar 

  81. Taylor, G.I.: Diffusion by continuous movements. Proc. Lond. Math. Soc. 2(20), 196–212 (1921)

    MathSciNet  MATH  Google Scholar 

  82. Trefry, M.G., Ruan, F.P., McLaughlin, D.: Numerical simulations of preasymptotic transport in heterogeneous porous media: departures from the Gaussian limit. Water Resour. Res. 39(3), 1063 (2003)

    Google Scholar 

  83. Vanderborght, J.: Concentration variance and spatial covariance in second order stationary heterogeneous conductivity fields. Water Resour. Res. 37(7), 1893–1912 (2001)

    Google Scholar 

  84. Yaglom, A.M.: Correlation Theory of Stationary and Related Random Functions, Volume I: Basic Results. Springer, New York (1987)

    MATH  Google Scholar 

  85. Zirbel, C.L.: Lagrangian observations of homogeneous random environments. Adv. Appl. Prob. 33, 810–835 (2001)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Friedrich-Alexander University of Erlangen-Nürnberg, Erlangen, Germany

    Nicolae Suciu

  2. Tiberiu Popoviciu Institute of Numerical Analysis, Cluj-Napoca Branch of the Romanian Academy, Cluj-Napoca, Romania

    Nicolae Suciu

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  1. Nicolae Suciu
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Suciu, N. (2019). Diffusion in Random Velocity Fields. In: Diffusion in Random Fields . Geosystems Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-15081-5_4

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