Anomalous dispersion in correlated porous media: a coupled continuous time random walk approach

Regular Article
Part of the following topical collections:
  1. Topical issue: Continuous Time Random Walk Still Trendy: Fifty-year History, Current State and Outlook


We study the causes of anomalous dispersion in Darcy-scale porous media characterized by spatially heterogeneous hydraulic properties. Spatial variability in hydraulic conductivity leads to spatial variability in the flow properties through Darcy’s law and thus impacts on solute and particle transport. We consider purely advective transport in heterogeneity scenarios characterized by broad distributions of heterogeneity length scales and point values. Particle transport is characterized in terms of the stochastic properties of equidistantly sampled Lagrangian velocities, which are determined by the flow and conductivity statistics. The persistence length scales of flow and transport velocities are imprinted in the spatial disorder and reflect the distribution of heterogeneity length scales. Particle transitions over the velocity length scales are kinematically coupled with the transition time through velocity. We show that the average particle motion follows a coupled continuous time random walk (CTRW), which is fully parameterized by the distribution of flow velocities and the medium geometry in terms of the heterogeneity length scales. The coupled CTRW provides a systematic framework for the investigation of the origins of anomalous dispersion in terms of heterogeneity correlation and the distribution of conductivity point values. We derive analytical expressions for the asymptotic scaling of the moments of the spatial particle distribution and first arrival time distribution (FATD), and perform numerical particle tracking simulations of the coupled CTRW to capture the full average transport behavior. Broad distributions of heterogeneity point values and lengths scales may lead to very similar dispersion behaviors in terms of the spatial variance. Their mechanisms, however are very different, which manifests in the distributions of particle positions and arrival times, which plays a central role for the prediction of the fate of dissolved substances in heterogeneous natural and engineered porous materials.


  1. 1.
    J.P. Bouchaud, A. Georges, Phys. Rep. 195, 127 (1990)ADSCrossRefMathSciNetGoogle Scholar
  2. 2.
    J. Klafter, I. Sokolov, Phys. World 18, 29 (2005)CrossRefGoogle Scholar
  3. 3.
    H. Scher, M. Lax, Phys. Rev. B 7, 4491 (1973)ADSCrossRefMathSciNetGoogle Scholar
  4. 4.
    H. Scher, E.W. Montroll, Phys. Rev. B 12, 2455 (1975)ADSCrossRefGoogle Scholar
  5. 5.
    M. Chevrollier, N. Mercadier, W. Guerin, R. Kaiser, Eur. Phys. J. D: At. Mol. Opt. Plasma Phys. 58, 161 (2010)CrossRefGoogle Scholar
  6. 6.
    P. Barthelemy, J. Bertolotti, D.S. Wiersma, Nature 453, 495 (2008)ADSCrossRefGoogle Scholar
  7. 7.
    G.M. Viswanathan, V. Afanasyev, S.V. Buldyrev, E.J. Murphy, P.E. Prince, H.E. Stanley, Nature 381, 413 (1996)ADSCrossRefGoogle Scholar
  8. 8.
    C. Brown, L.S. Liebovitch, R. Glendon, Human Ecol. 35, 129 (2006)CrossRefGoogle Scholar
  9. 9.
    S.R. Yu, M. Burkhardt, M. Nowak, J. Ries, Z. Petrášek, S. Scholpp, P. Schwille, M. Brand, Nature 461, 533 (2009)ADSCrossRefGoogle Scholar
  10. 10.
    J.-H. Jeon, M. Javanainen, H. Martinez-Seara, R. Metzler, I. Vattulainen, Phys. Rev. X 6, 021006 (2016)Google Scholar
  11. 11.
    P. Massignan, C. Manzo, J. Torreno-Pina, M. Garcia-Parajo, M. Lewenstein, J.G. Lapeyre, Phys. Rev. Lett. 112, 150603 (2014)ADSCrossRefGoogle Scholar
  12. 12.
    B. Berkowitz, A. Cortis, M. Dentz, H. Scher, Rev. Geophys. 44, RG2003 (2006)ADSCrossRefGoogle Scholar
  13. 13.
    J. Bear, Dynamics of fluids in porous media (American Elsevier, New York, 1972)Google Scholar
  14. 14.
    H. Brenner, D. Edwards, Macrotransport processes (Butterworth-Heinemann, MA, USA, 1993)Google Scholar
  15. 15.
    L.W. Gelhar, C. Welty, K.R. Rehfeldt, Water Resour. Res. 28, 1955 (1992)ADSCrossRefGoogle Scholar
  16. 16.
    J.H. Cushman, T.R. Ginn, Transp. Porous Media 13, 123 (1993)CrossRefGoogle Scholar
  17. 17.
    R. Haggerty, S.M. Gorelick, Water Resour. Res. 31, 2383 (1995)ADSCrossRefGoogle Scholar
  18. 18.
    B. Berkowitz, H. Scher, Water Resour. Res. 31, 1461 (1995)ADSCrossRefGoogle Scholar
  19. 19.
    V. Cvetkovic, H. Cheng, X.-H. Wen, Water Resour. Res. 32, 1671 (1996)ADSCrossRefGoogle Scholar
  20. 20.
    B. Berkowitz, H. Scher, Phys. Rev. Lett. 79, 4038 (1997)ADSCrossRefGoogle Scholar
  21. 21.
    J. Carrera, X. Sánchez-Vila, I. Benet, A. Medina, G. Galarza, J. Guimerà, Hydrogeol. J. 6, 178 (1998)ADSCrossRefGoogle Scholar
  22. 22.
    R. Haggerty, S.A. McKenna, L.C. Meigs, Water Resour. Res. 36, 3467 (2000)ADSCrossRefGoogle Scholar
  23. 23.
    M. Willmann, J. Carrera, X. Sanchez-Vila, Water Resour. Res. 44, W12437 (2008)ADSCrossRefGoogle Scholar
  24. 24.
    T. Le Borgne, M. Dentz, J. Carrera, Phys. Rev. Lett. 101, 090601 (2008)ADSCrossRefGoogle Scholar
  25. 25.
    V. Cvetkovic, A. Fiori, G. Dagan, Water Resour. Res. 50, WR015449 (2014)Google Scholar
  26. 26.
    G. Dagan, J. Fluid Mech. 145, 151 (1984)ADSCrossRefGoogle Scholar
  27. 27.
    L.W. Gelhar, C.L. Axness, Water Resour. Res. 19, 161 (1983)ADSCrossRefGoogle Scholar
  28. 28.
    Y. Rubin, Applied stochastic hydrogeology (Oxford University Press, New York, 2003)Google Scholar
  29. 29.
    J.H. Cushman, X. Hu, T.R. Ginn, J. Stat. Phys. 75, 859 (1994)ADSCrossRefGoogle Scholar
  30. 30.
    A. Guadagnini, S.P. Neuman, Water Resour. Res. 35, 2999 (1999)ADSCrossRefGoogle Scholar
  31. 31.
    M.M. Meerschaert, D.A. Benson, B. Baumer, Phys. Rev. E 59, 5026 (1999)ADSCrossRefGoogle Scholar
  32. 32.
    D.A. Benson, S.W. Wheatcraft, M.M. Meerschaert, Water Resour. Res. 36, 1403 (2000)ADSCrossRefGoogle Scholar
  33. 33.
    D. Zhang, D.A. Benson, Geophys. Res. Lett. 35, L07403 (2008)ADSCrossRefGoogle Scholar
  34. 34.
    D.A. Benson, M.M. Meerschaert, J. Revielle, Adv. Water Resour. 51, 479 (2013)ADSCrossRefGoogle Scholar
  35. 35.
    F. Delay, P. Ackerer, C. Danquigny, Vadose Zone J. 4, 360 (2005)CrossRefGoogle Scholar
  36. 36.
    R. Benke, S. Painter, Water Resour. Res. 39, 1324 (2003)ADSCrossRefGoogle Scholar
  37. 37.
    A. Fiori, I. Jankovic, G. Dagan, V. Cvetkovic, Water Resour. Res. 43, W09407 (2007)ADSGoogle Scholar
  38. 38.
    M. Dentz, T. Le Borgne, A. Englert, B. Bijeljic, J. Contam. Hydrol. 120, 1 (2011)ADSCrossRefGoogle Scholar
  39. 39.
    A. Russian, M. Dentz, P. Gouze, Water Resour. Res. 52, 3309 (2016)ADSCrossRefGoogle Scholar
  40. 40.
    B. Noetinger, D. Roubinet, A. Russian, T. Le Borgne, F. Delay, M. Dentz, J.-R. De Dreuzy, P. Gouze, Transp. Porous Media 1 (2016)Google Scholar
  41. 41.
    Y. Hatano, N. Hatano, Water Resour. Res. 34, 1027 (1998)ADSCrossRefGoogle Scholar
  42. 42.
    M. Dentz, A. Cortis, H. Scher, B. Berkowitz, Adv. Water Resour. 27, 155 (2004)ADSCrossRefGoogle Scholar
  43. 43.
    E.W. Montroll, G.H. Weiss, J. Math. Phys. 6, 167 (1965)ADSCrossRefGoogle Scholar
  44. 44.
    M. Dentz, A. Russian, P. Gouze, Phys. Rev. E 93, 010101(R) (2016)ADSCrossRefGoogle Scholar
  45. 45.
    R. Metzler, J. Klafter, Phys. Rep. 339, 1 (2000)ADSCrossRefGoogle Scholar
  46. 46.
    E. Barkai, Y. Garini, R. Metzler, Phys. Today 8, 29 (2012)CrossRefGoogle Scholar
  47. 47.
    R. Kutner, J. Masoliver, Eur. Phys. J. B 90, 50 (2017)ADSCrossRefGoogle Scholar
  48. 48.
    M. Shlesinger, Eur. Phys. J. B 90, 93 (2017)ADSCrossRefGoogle Scholar
  49. 49.
    P. Saffman, J. Fluid Mech. 6, 321 (1959)ADSCrossRefMathSciNetGoogle Scholar
  50. 50.
    B. Bijeljic, M.J. Blunt, Water Resour. Res. 42, W01202 (2006)ADSCrossRefGoogle Scholar
  51. 51.
    T. Le Borgne, D. Bolster, M. Dentz, P. de Anna, A.M. Tartakovsky, Water Resour. Res. 47, W12538 (2011)ADSCrossRefGoogle Scholar
  52. 52.
    B. Bijeljic, P. Mastaghimi, M.J. Blunt, Phys. Rev. Lett. 107, 204502 (2011)ADSCrossRefGoogle Scholar
  53. 53.
    B. Bijeljic, A. Raeini, P. Mostaghimi, M.J. Blunt, Phys. Rev. E 87, 013011 (2013)ADSCrossRefGoogle Scholar
  54. 54.
    P. De Anna, T. Le Borgne, M. Dentz, A.M. Tartakovsky, D. Bolster, P. Davy, Phys. Rev. Lett. 110, 184502 (2013)ADSCrossRefGoogle Scholar
  55. 55.
    P.K. Kang, P. de Anna, J. Nunes, B. Bijeljic, M.J. Blunt, R. Juanes, Geophys. Res. Lett. 41, 6184 (2014)ADSCrossRefGoogle Scholar
  56. 56.
    F. Gjetvaj, A. Russian, P. Gouze, M. Dentz, Water Resour. Res. 51, 8273 (2015)ADSCrossRefGoogle Scholar
  57. 57.
    M. Holzner, V.L. Morales, M. Willmann, M. Dentz, Phys. Rev. E 92, 013015 (2015)ADSCrossRefGoogle Scholar
  58. 58.
    M. Dentz, P.K. Kang, A. Comolli, T. Le Borgne, D.R. Lester, Phys. Rev. Fluids 1, 074004 (2016)ADSCrossRefGoogle Scholar
  59. 59.
    P.K. Kang, T. Le Borgne, M. Dentz, O. Bour, R. Juanes, Water Resour. Res. 51, 940 (2015)ADSCrossRefGoogle Scholar
  60. 60.
    M. Dentz, P.K. Kang, T. Le Borgne, Adv. Water Res. 82, 16 (2015)CrossRefGoogle Scholar
  61. 61.
    T. Le Borgne, M. Dentz, J. Carrera, Phys. Rev. E 78, 041110 (2008)CrossRefGoogle Scholar
  62. 62.
    P.K. Kang, M. Dentz, T. Le Borgne, R. Juanes, Phys. Rev. Lett. 107, 180602 (2011)ADSCrossRefGoogle Scholar
  63. 63.
    P.K. Kang, M. Dentz, T. Le Borgne, S. Lee, R. Juanes, Adv. Water Resour. 106, 80 (2017)ADSCrossRefGoogle Scholar
  64. 64.
    M. Dentz, B. Berkowitz, Water Resour. Res. 39, 1111 (2003)ADSCrossRefGoogle Scholar
  65. 65.
    G. Margolin, M. Dentz, B. Berkowitz, Chem. Phys. 295, 71 (2003)ADSCrossRefGoogle Scholar
  66. 66.
    M. Dentz, B. Berkowitz, Phys. Rev. E 72, 031110 (2005)ADSCrossRefMathSciNetGoogle Scholar
  67. 67.
    M. Dentz, A. Castro, Geophys. Res. Lett. 36, L03403 (2009)ADSCrossRefGoogle Scholar
  68. 68.
    D.A. Benson, M.M. Meerschaert, Adv. Water Res. 32, 532 (2009)CrossRefGoogle Scholar
  69. 69.
    M. Dentz, P. Gouze, A. Russian, J. Dweik, F. Delay, Adv. Water Res. 49, 13 (2012)CrossRefGoogle Scholar
  70. 70.
    A. Comolli, J.J. Hidalgo, C. Moussey, M. Dentz, Transp. Porous Media 1 (2016)Google Scholar
  71. 71.
    M. Dentz, D. Bolster, Phys. Rev. Lett. 105, 244301 (2010)ADSCrossRefGoogle Scholar
  72. 72.
    A. Tyukhova, M. Dentz, W. Kinzelbach, M. Willmann, Phys. Rev. Fluids 1, 074002 (2016)ADSCrossRefGoogle Scholar
  73. 73.
    G. Christakos, Random field models in earth sciences (Academic Press, 1992)Google Scholar
  74. 74.
    I. Eames, J.W.M. Bush, Proc. R. Soc. Lond. A 455, 3665 (1999)ADSCrossRefGoogle Scholar
  75. 75.
    Y. Edery, A. Guadagnini, H. Scher, B. Berkowitz, Water Resour. Res. 50, 1490 (2014)ADSCrossRefGoogle Scholar
  76. 76.
    V. Hakoun, A. Comolli, M. Dentz, Unpublished, 2017Google Scholar
  77. 77.
    M. Dentz, D.R. Lester, T.L. Borgne, F.P.J. de Barros, Phys. Rev. E 94, 061102(R) (2016)ADSCrossRefGoogle Scholar
  78. 78.
    M. Abramowitz, I.A. Stegun, Handbook of mathematical functions (Dover publications, New York, 1972)Google Scholar
  79. 79.
    S. Attinger, M. Dentz, W. Kinzelbach, Stoch. Environ. Res. Risk Assess. 18, 9 (2004)CrossRefGoogle Scholar
  80. 80.
    M. Dentz, H. Scher, D. Holder, B. Berkowitz, Phys. Rev. E 78, 041110 (2008)ADSCrossRefGoogle Scholar
  81. 81.
    M.F. Shlesinger, B.J. West, J. Klafter, Phys. Rev. Lett. 58, 1100 (1987)ADSCrossRefMathSciNetGoogle Scholar
  82. 82.
    J. Klafter, A. Blumen, G. Zumofen, M.F. Shlesinger, Physica A 168, 637 (1990)ADSCrossRefGoogle Scholar
  83. 83.
    M.M. Meerschaert, E. Nane, Y. Xiao, Stat. Prob. Lett. 79, 1194 (2009)CrossRefGoogle Scholar
  84. 84.
    A. Rebenshtok, S. Denisov, P. Hänggi, E. Barkai, Phys. Rev. E 90, 062135 (2014)ADSCrossRefGoogle Scholar
  85. 85.
    V. Zaburdaev, S. Denisov, J. Klafter, Rev. Mod. Phys. 87, 483 (2015)ADSCrossRefGoogle Scholar
  86. 86.
    M. Dentz, T. Le Borgne, D.R. Lester, F.P.J. de Barros, Phys. Rev. E 92, 032128 (2015)ADSCrossRefGoogle Scholar
  87. 87.
    B. Berkowitz, J. Klafter, R. Metzler, H. Scher, Water Resour. Res. 38, 1191 (2002)ADSCrossRefGoogle Scholar
  88. 88.
    J. Klafter, R. Silbey, Phys. Rev. Lett. 44, 55 (1980)ADSCrossRefGoogle Scholar
  89. 89.
    M.F. Shlesinger, J. Klafter, Y.M. Wong, J. Stat. Phys. 27, 499 (1982)ADSCrossRefGoogle Scholar
  90. 90.
    M. Shlesinger, J. Stat. Phys. 10, 421 (1974)ADSCrossRefGoogle Scholar
  91. 91.
    G. Margolin, B. Berkowitz, Phys. Rev. E 65, 1 (2002)CrossRefGoogle Scholar

Copyright information

© EDP Sciences, SIF, Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Institute of Environmental Assessment and Water Research (IDAEA-CSIC)BarcelonaSpain
  2. 2.Department of Civil and Environmental EngineeringTechnical University of Catalonia (UPC)BarcelonaSpain
  3. 3.Associated Unit: Hydrogeology Group (UPC-CSIC)BarcelonaSpain

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