Transport in Porous Media

, Volume 115, Issue 2, pp 345–385 | Cite as

Random Walk Methods for Modeling Hydrodynamic Transport in Porous and Fractured Media from Pore to Reservoir Scale

  • Benoit Noetinger
  • Delphine Roubinet
  • Anna Russian
  • Tanguy Le Borgne
  • Frederick Delay
  • Marco Dentz
  • Jean-Raynald de Dreuzy
  • Philippe Gouze


Random walk (RW) methods are recurring Monte Carlo methods used to model convective and diffusive transport in complex heterogeneous media. Many applications can be found, including fluid mechanic, hydrology and chemical reactors modeling. These methods are easy to implement, very versatile and flexible enough to become appealing for many applications because they generally overlook or deeply simplify the building of explicit complex meshes required by deterministic methods. RW provides a good physical understanding of the interactions between the space scales of heterogeneities and the transport phenomena under consideration. In addition, they can result in efficient upscaling methods, especially in the context of flow and transport in fractured media. In the present study, we review the applications of RW to several situations that cope with diverse spatial scales and different insights into upscaling problems. The advantages and downsides of RW are also discussed, thus providing a few avenues for further works and applications.


Random walk Random media Fractured media Diffusion Dispersion Upscaling Transfers Multiple porosity 


  1. Aanonsen, S.I., Nævdal, G., Oliver, D.S., Reynolds, A.C., Vallès, B., et al.: The ensemble Kalman filter in reservoir engineering—a review. Spe J. 14(03), 393–412 (2009)CrossRefGoogle Scholar
  2. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions. Dover Publications, New York (1972)Google Scholar
  3. Acuna, J.A., Yortsos, Y.C.: Application of fractal geometry to the study of networks of fractures and their pressure transient. Water Resour. Res. 31(3), 527–540 (1995). doi: 10.1029/94WR02260 CrossRefGoogle Scholar
  4. Arbogast, T., Douglas, J., Hornung, U.: Derivation of the double porosity model of single phase flow via homogeneization theory. SIAM J. Math. Anal. 21(4), 823–836 (1990)CrossRefGoogle Scholar
  5. Aris, R.: On the dispersion of a solute in a fluid flowing through a tube. Proc. R. Soc. Lond. A Math. Phys. Sci. 235, 67–77 (1956). doi: 10.1098/rspa.1956.0065 CrossRefGoogle Scholar
  6. Babey, T., de Dreuzy, J.-R., Casenave, C.: Multi-rate mass transfer (MRMT) models for general diffusive porosity structures. Adv. Water Res. 76, 146–156 (2015). doi: 10.1016/j.advwatres.2014.12.006 CrossRefGoogle Scholar
  7. Barenblatt, G.I., Zheltov, Y.P.: Fundamental equations of homogeneous liquids in fissured rocks. Dokl Akad Nauk SSSR 132(3), 545–548 (1960)Google Scholar
  8. Barkai, E., Garini, Y., Metzler, R.: Strange kinetics of single molecules in living cells. Phys. Today 65(8), 29–35 (2012). doi: 10.1063/PT.3.1677 CrossRefGoogle Scholar
  9. Barker, J.A.: A generalized radial flow model for hydraulic tests in fractured rock. Water Resour. Res. 24(10), 1796–1804 (1988). doi: 10.1029/WR024i010p01796 CrossRefGoogle Scholar
  10. Barthelemy, P., Bertolotti, J., Wiersma, D.S.: A Lévy flight for light. Nature 453(7194), 495–498 (2008). doi: 10.1038/nature06948 CrossRefGoogle Scholar
  11. Bear, J.: Dynamics of Fluids in Porous Media. Dover Publications, Mineola (1973)Google Scholar
  12. Beaudoin, A., de Dreuzy, J.R.: Numerical assessment of 3-D macrodispersion in heterogeneous porous media. Water Resour. Res. 49(5), 2489–2496 (2013). doi: 10.1002/wrcr.20206 CrossRefGoogle Scholar
  13. Beaudoin, A., Huberson, S., Rivoalen, E.: Anisotropic particle method. C. R. Mec. 330(1), 51–56 (2002). doi: 10.1016/S1631-0721(02)01429-8 CrossRefGoogle Scholar
  14. Beaudoin, A., Huberson, S., Rivoalen, E.: Simulation of anisotropic diffusion by means of a diffusion velocity method. J. Comput. Phys. 186(1), 122–135 (2003). doi: 10.1016/S0021-991(03)00024-X CrossRefGoogle Scholar
  15. Beaudoin, A., de Dreuzy, J.R., Erhel, J.: An efficient parallel tracker for advection-diffusion simulations in heterogeneous porous media. In: Kermarrec, A.-M., Bougé, L., Priol, T. (eds.) Europar, pp. 28–31. Springer, Heidelberg (2007)Google Scholar
  16. Beaudoin, A., de Dreuzy, J.R., Erhel, J.: Numerical Monte Carlo analysis of the influence of pore-scale dispersion on macrodispersion in 2-D heterogeneous porous media. Water Resour. Res. 46, 12 (2010). doi: 10.1029/2010WR009576 CrossRefGoogle Scholar
  17. Bechtold, M., Vanderborght, J., Ippisch, O., Vereecken, H.: Efficient random walk particle tracking algorithm for advective-dispersive transport in media with discontinuous dispersion coefficients and water contents. Water Resour. Res. 47, 10 (2011). doi: 10.1029/2010WR010267 CrossRefGoogle Scholar
  18. Becker, M.W., Shapiro, A.M.: Interpreting tracer breakthrough tailing from different forced-gradient tracer experiment configurations in fractured bedrock. Water Resour. Res. 39(1), 1024 (2003). doi: 10.1029/2001WR001190 CrossRefGoogle Scholar
  19. Bel, G., Barkai, E.: Weak ergodicity breaking in the continuous-time random walk. Phys. Rev. Lett. 94(240), 602 (2005). doi: 10.1103/PhysRevLett.94.240602
  20. Berkowitz, B., Balberg, I.: Percolation theory and its application to groundwater hydrology. Water Resour. Res. 29(4), 775–794 (1993). doi: 10.1029/92WR02707 CrossRefGoogle Scholar
  21. Berkowitz, B., Scher, H.: Anomalous transport in random fracture networks. Phys. Rev. Lett. 79(20), 4038–4041 (1997). doi: 10.1103/PhysRevLett.79.4038
  22. Berkowitz, B., Scher, H.: Theory of anomalous chemical transport in random fracture networks. Phys. Rev. E 57(5), 5858–5869 (1998). doi: 10.1103/PhysRevE.57.5858 CrossRefGoogle Scholar
  23. Berkowitz, B., Naumann, C., Smith, L.: Mass-transfer at fracture intersections - An evaluation of mixing models. Water Resour. Res. 30(6), 1765–1773 (1994). doi: 10.1029/94WR00432 CrossRefGoogle Scholar
  24. Berkowitz, B., Scher, H., Silliman, S.: Anomalous transport in laboratory-scale, heterogeneous porous media. Water Resour. Res. 36(1), 149–158 (2000). doi: 10.1029/1999WR900295 CrossRefGoogle Scholar
  25. Berkowitz, B., Klafter, J., Metzler, R., Scher, H.: Physical pictures of transport in heterogeneous media: advection-dispersion, random-walk, and fractional derivative formulations. Water Resour. Res. 38(10), 1191 (2002). doi: 10.1029/2001WR001030 CrossRefGoogle Scholar
  26. Berkowitz, B., Cortis, A., Dentz, M., Scher, H.: Modeling non-Fickian transport in geological formations as a continuous time random walk. Rev. Geophys. 44(2), RG2003 (2006). doi: 10.1029/2005RG000178 CrossRefGoogle Scholar
  27. Besnard, K., de Dreuzy, J.R., Davy, P., Aquilina, L.: A modified Lagrangian-volumes method to simulate nonlinearly and kinetically sorbing solute transport in heterogeneous porous media. J. Contam. Hydrol. 120–21(SI), 89–98 (2011). doi: 10.1016/j.jconhyd.2010.03.004 CrossRefGoogle Scholar
  28. Bijeljic, B., Mostaghimi, P., Blunt, M.: Insights into non-Fickian solute transport in carbonates. Water Resour. Res. 49(5), 2714–2728 (2013a)CrossRefGoogle Scholar
  29. Bijeljic, B., Raeini, A., Mostaghimi, P., Blunt, M.: Predictions of non-fickian solute transport in different classes of porous media using direct simulation on pore-scale images. Phys. Rev. E 87(1), 013011 (2013b). doi: 10.1103/PhysRevE.87.013011 CrossRefGoogle Scholar
  30. Boano, F., Packman, A.I., Cortis, A., Revelli, R., Ridolfi, L.: A continuous time random walk approach to the stream transport of solutes. Water Resour. Res. 43(10) (2007). doi: 10.1029/2007WR006062
  31. Bodin, J.: From analytical solutions of solute transport equations to multidimensional time-domain random walk (TDRW) algorithms. Water Resour. Res. 51(3), 1860–1871 (2015). doi: 10.1002/2014WR015910 CrossRefGoogle Scholar
  32. Bodin, J., Porel, G., Delay, F.: Simulation of solute transport in discrete fracture networks using the time domain random walk method. Earth Planet. Sci. Lett. 208(3–4), 297–304 (2003). doi: 10.1016/S0012-821X(03)00052-9 CrossRefGoogle Scholar
  33. Bodin, J., Porel, G., Delay, F., Ubertosi, F., Bernard, S., de Dreuzy, J.R.: Simulation and analysis of solute transport in 2D fracture/pipe networks: the SOLFRAC program. J. Contam. Hydrol. 89(1–2), 1–28 (2007). doi: 10.1016/j.jconhyd.2006.07.005 CrossRefGoogle Scholar
  34. Bouchaud, J.P., Georges, A.: Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys. Rep. 195(4–5), 127–293 (1990). doi: 10.1016/0370-1573(90)90099-N CrossRefGoogle Scholar
  35. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, Berlin (1991)CrossRefGoogle Scholar
  36. Bromly, M., Hinz, C.: Non-Fickian transport in homogeneous unsaturated repacked sand. Water Resour. Res. 40(7) (2004). doi: 10.1029/2003WR002579
  37. Bruderer, C., Bernabé, Y.: Network modeling of dispersion: transition from Taylor dispersion in homogeneous networks to mechanical dispersion in very heterogeneous ones. Water Resour. Res. 37(4), 897–908 (2001). doi: 10.1029/2000WR900362 CrossRefGoogle Scholar
  38. Cacas, M.C., Ledoux, E., de Marsily, G., Barbreau, A., Calmels, P., Gaillard, B., Margritta, R.: Modeling fracture flow with a stochastic discrete network: calibration and validation. 2. The transport model. Water Resour. Res. 26(3), 491–500 (1990). doi: 10.1029/WR026i003p00491 Google Scholar
  39. Carrera, J., Sanchez-Vila, X., Benet, I., Medina, A., Galarza, G., Guimera, J.: On matrix diffusion: formulations, solution methods and qualitative effects. Hydrogeol. J. 6(1), 178–190 (1998). doi: 10.1007/s100400050143 CrossRefGoogle Scholar
  40. Carslaw, H.S., Jaeger, J.C.: Conduction of heat in solids. Oxford science publications, Clarendon Press, Oxford (1986)Google Scholar
  41. Chang, J., Yortsos, Y.C.: Pressure transient analysis of fractal reservoirs. SPE Form. Eval. 5(1) (1990). doi: 10.2118/18170-PA
  42. Charlaix, E., Guyon, E., Roux, S.: Permeability of a random array of fractures of widely varying apertures. Transp. Porous Media 2(1), 31–43 (1987)CrossRefGoogle Scholar
  43. Chavent, G., Roberts, J.E.: A unified physical presentation of mixed, mixed-hybrid finite elements and standard finite difference approximations for the determination of velocities in waterflow problems. Adv. Water Res. 14(6), 329–348 (1991). doi: 10.1016/0309-1708(91)90020-O CrossRefGoogle Scholar
  44. Chen, Z.X.: Transient Flow of Slightly Compressible Fluids Through Double-porosity, Double-permeability systems–A state-of-the-art review. Transp. Porous Media 4(2), 147–184 (1989). doi: 10.1007/BF00134995 CrossRefGoogle Scholar
  45. Cirpka, O.A.: Effects of sorption on transverse mixing in transient flows. J. Contam. Hydrol. 78(3), 207–229 (2005). doi: 10.1016/j.jconhyd.2005.05.008 CrossRefGoogle Scholar
  46. Cordes, C., Kinzelbach, W.: Continuous groundwater velocity field and path lines in linear, bilinear and trilinear finite elements. Water Resour. Res. 28(11), 2903–2911 (1992). doi: 10.1029/92WR01686 CrossRefGoogle Scholar
  47. Cordes, C., Kinzelbach, W.: Comment on “Application of the mixed hybrid finite element approximation in a groundwater flow model: Luxury or necessity?”. Water Resour. Res. 32(6), 1905–1911 (1996). doi: 10.1029/96WR00567 CrossRefGoogle Scholar
  48. Cortis, A., Berkowitz, B.: Anomalous transport in “classical” soil and sand columns. Soil Sci. Soc. Am. J. 68(5), 1539–1548 (2004)CrossRefGoogle Scholar
  49. Cortis, A., Ghezzehei, T.A.: On the transport of emulsions in porous media. J. Colloid Interface Sci. 313(1), 1–4 (2007). doi: 10.1016/j.jcis.2007.04.021 CrossRefGoogle Scholar
  50. Cortis, A., Knudby, C.: A continuous time random walk approach to transient flow in heterogeneous porous media. Water Resour. Res. (2006). doi: 10.1029/2006WR005227 Google Scholar
  51. Cvetkovic, V., Frampton, A.: Solute transport and retention in three-dimensional fracture networks. Water Resour. Res. (2012). doi: 10.1029/2011WR011086 Google Scholar
  52. Cvetkovic, V., Painter, S., Outters, N., Selroos, J.O.: Stochastic simulation of radionuclide migration in discretely fractured rock near the Äspö Hard Rock Laboratory. Water Resour. Res. (2004). doi: 10.1029/2003WR002655 Google Scholar
  53. Dagan, G.: Flow and Transport in Porous Formations. Springer, Berlin (1989)CrossRefGoogle Scholar
  54. Danckwerts, P.V.: The definition and measurements of some characteristics of mixtures. Appl. Sci. Res. 3(4), 279–296 (1952)Google Scholar
  55. Daviau, F.: Interprétation des essais de puits, les méthodes nouvelles, technip edn. Publications de l’institut francais du pétrole, Paris (1986)Google Scholar
  56. de Anna, P., Le Borgne, T., Dentz, M., Tartakovsky, A., Bolster, D., Davy, P.: Flow intermittency, dispersion, and correlated continuous time random walks in porous media. Phys. Rev. Lett. 110(18), 184502 (2013). doi: 10.1103/PhysRevLett.110.184502
  57. de Arcangelis, L., Koplik, J., Redner, S., Wilkinson, D.: Hydrodynamic dispersion in network models of porous media. Phys. Rev. Lett. 57(8), 986–999 (1986). doi: 10.1103/PhysRevLett.57.996
  58. de Simoni, M., Carrera, J., Sanchez-Vila, X., Guadagnini, A.: A procedure for the solution of multicomponent reactive transport problems. Water Resour. Res. 41(11), (2005). doi: 10.1029/2005WR004056
  59. de Swaan, A.: Analytic solutions for determining naturally fractured reservoir properties by well testing. SPE J. 16(3), 117–22 (1976)CrossRefGoogle Scholar
  60. de Swann, A., Ramirez-Villa, M.: Functions of flow from porous rock blocks. J. Petrol. Sci. Eng. 9(1), 39–48 (1993). doi: 10.1016/0920-4105(93)90027-C CrossRefGoogle Scholar
  61. Delay, F., Bodin, J.: Time domain random walk method to simulate transport by advection-dispersion and matrix diffusion in fractured networks. Geophys. Res. Lett. 28(21), 4051–4054 (2001). doi: 10.1029/2001GL013698 CrossRefGoogle Scholar
  62. Delay, F., Porel, G., Sardini, P.: Modelling diffusion in a heterogeneous rock matrix with a time-domain Lagrangian method and an inversion procedure. C. R. Geosci. 334(13), 967–973 (2002). doi: 10.1016/S1631-0713(02)01835-7 CrossRefGoogle Scholar
  63. Delay, F., Ackerer, P., Danquigny, C.: Simulating solute transport in porous or fractured formations using random walk particle tracking: a review. Vadose Zone J. 4(2), 360–379 (2005). doi: 10.2136/vzj2004.0125 CrossRefGoogle Scholar
  64. Delorme, M., Daniel, J.M., Kada-Kloucha, C., Khvoenkova, N., Schueller, S., Souque, C.: An efficient model to simulate reservoir stimulation and induced microseismic events on 3D discrete fracture network for unconventional reservoirs. In: Unconventional Resources Technology Conference, 12–14 August, Denver, Colorado, USA, pp 1433–1442, doi: 10.1190/URTEC2013-146 (2013a)
  65. Delorme, M., Mota, R.O., Khvoenkova, N., Fourno, A., Noetinger, B.: A methodology to characterize fractured reservoirs constrained by statistical geological analysis and production: a real field case study. Geol. Soc. Lond. Special Publ. 374(1), 273–288 (2013b)CrossRefGoogle Scholar
  66. Dentz, M., Cortis, A., Scher, H., Berkowitz, B.: Time behavior of solute transport in heterogeneous media: transition from anomalous to normal transport. Adv. Water Resour. 27(2), 155–173 (2004). doi: 10.1016/j.advwatres.2003.11.002 CrossRefGoogle Scholar
  67. Dentz, M., Le Borgne, T., Englert, A., Bijeljic, B.: Mixing, spreading and reaction in heterogeneous media: a brief review. J. Contam. Hydrol. 120–21(SI), 1–17 (2011). doi: 10.1016/j.jconhyd.2010.05.002 CrossRefGoogle Scholar
  68. Dentz, M., Gouze, P., Russian, A., Dweik, J., Delay, F.: Diffusion and trapping in heterogeneous media: an inhomogeneous continuous time random walk approach. Adv. Water Resour. 49, 13–22 (2012). doi: 10.1016/j.advwatres.2012.07.015 CrossRefGoogle Scholar
  69. Dentz, M., Russian, A., Gouze, P.: Self-averaging and ergodicity of subdiffusion in quenched random media. Phys. Rev. E 93(1), 010101 (2016)Google Scholar
  70. Dershowitz, W., Miller, I.: Dual porosity fracture flow and transport. Geophys. Res. Lett. 22(11), 1441–1444 (1995). doi: 10.1029/95GL01099 CrossRefGoogle Scholar
  71. de Dreuzy, J.R., Davy, P., Berkowitz, B.: Advective transport in the percolation backbone in two dimensions. Phys. Rev. E 64(5), 1–4 (2001)Google Scholar
  72. de Dreuzy, J.R., Beaudoin, A., Erhel, J.: Asymptotic dispersion in 2D heterogeneous porous media determined by parallel numerical simulations. Water Resour. Res. (2007). doi: 10.1029/2006WR005394 Google Scholar
  73. de Dreuzy, J.R., Carrera, J., Dentz, M., Le Borgne, T.: Time evolution of mixing in heterogeneous porous media. Water Resour. Res. (2012). doi: 10.1029/2011WR011360 Google Scholar
  74. de Dreuzy, J.R., Rapaport, A., Babey, T., Harmand, J.: Influence of porosity structures on mixing-induced reactivity at chemical equilibrium in mobile/immobile Multi-Rate Mass Transfer (MRMT) and Multiple INteracting Continua (MINC) models. Water Resour. Res. 49(12), 8511–8530 (2013). doi: 10.1002/2013WR013808 CrossRefGoogle Scholar
  75. Edery, Y., Guadagnini, A., Scher, H., Berkowitz, B.: Origins of anomalous transport in heterogeneous media: Structural and dynamic controls. Water Resour. Res. 50(2), 1490–1505 (2014). doi: 10.1002/2013WR015111 CrossRefGoogle Scholar
  76. Einstein, A.: Investigations on the theory of the Brownian movement. Dover Publication, New York (1956)Google Scholar
  77. Emmanuel, S., Berkowitz, B.: Continuous time random walks and heat transfer in porous media. Transp. Porous Media 67(3), 413–430 (2007). doi: 10.1007/s11242-006-9033-z CrossRefGoogle Scholar
  78. Evensen, G.: Data assimilation: the ensemble Kalman filter. Springer, Berlin (2009)CrossRefGoogle Scholar
  79. Fernàndez-Garcia, D., Sanchez-Vila, X.: Optimal reconstruction of concentrations, gradients and reaction rates from particle distributions. J. Contam. Hydrol. 120, 99–114 (2011)Google Scholar
  80. Fleury, M., Bauer, D., Néel, M.: Modeling of super-dispersion in unsaturated porous media using NMR propagators. Microporous Mesoporous Mater. 205, 75–78 (2015)CrossRefGoogle Scholar
  81. Geiger, S., Cortis, A., Birkholzer, J.T.: Upscaling solute transport in naturally fractured porous media with the continuous time random walk method. Water Resour. Res. 46, 1–13 (2010). doi: 10.1029/2010WR009133
  82. Gjetvaj, F., Russian, A., Gouze, P., Dentz, M.: Dual control of flow field heterogeneity and immobile porosity on non-Fickian transport in Berea sandstone. Water Resour. Res. 51(10), 8273–8293 (2015). doi: 10.1002/2015WR017645 CrossRefGoogle Scholar
  83. Gouze, P., Luquot, L.: X-ray microtomography characterization of porosity, permeability and reactive surface changes during dissolution. J. Contam. Hydrol. 120–21(SI), 45–55 (2011). doi: 10.1016/j.jconhyd.2010.07.004 CrossRefGoogle Scholar
  84. Gouze, P., Le Borgne T., Leprovost, R., Lods, G., Poidras, T., Pezard, P.: Non-Fickian dispersion in porous media: 1. Multiscale measurements using single-well injection withdrawal tracer tests. Water Resour. Res. (2008a). doi: 10.1029/2007WR006278
  85. Gouze, P., Melean, Y., Le Borgne, T., Dentz, M., Carrera, J.: Non-fickian dispersion in porous media explained by heterogeneous microscale matrix diffusion. Water Resour. Res. (2008b). doi: 10.1029/2007WR006690 Google Scholar
  86. Guillon, V., Fleury, M., Bauer, D., Néel, M.C.: Superdispersion in homogeneous unsaturated porous media using NMR propagators. Phys. Rev. E (2013). doi: 10.1103/PhysRevE.87.043007 Google Scholar
  87. Guillon, V., Bauer, D., Fleury, M., Néel, M.C.: Computing the longtime behaviour of NMR propagators in porous media using a pore network random walk model. Transp. Porous Media 101(2), 251–267 (2014). doi: 10.1007/s11242-013-0243-x CrossRefGoogle Scholar
  88. Haggerty, R., Gorelick, S.M.: Multiple-rate mass transfer for modeling diffusion and surface reactions in media with pore-scale heterogeneity. Water Resour. Res. 31(10), 2383–2400 (1995). doi: 10.1029/95WR01583 CrossRefGoogle Scholar
  89. Haggerty, R., McKenna, S.A., Meigs, L.C.: On the late time behavior of tracer test breakthrough curves. Water Resour. Res. 36(12), 3467–3479 (2000). doi: 10.1029/2000WR900214 CrossRefGoogle Scholar
  90. Hatano, Y., Hatano, N.: Dispersive transport of ions in column experiments: an explanation of long-tailed profiles. Water Resour. Res. 34(5), 1027–1033 (1998). doi: 10.1029/98WR00214 CrossRefGoogle Scholar
  91. He, Y., Burov, S., Metzler, R., Barkai, E.: Random time-scale invariant diffusion and transport coefficients. Phys. Rev. Lett. (2008). doi: 10.1103/PhysRevLett.101.058101
  92. Herrera, P.A., Beckie, R.D.: An assessment of particle methods for approximating anisotropic dispersion. Int. J. Numer. Methods Fluids 71(5), 634–651 (2013). doi: 10.1002/fld.3676 CrossRefGoogle Scholar
  93. Herrera, P.A., Massabo, M., Beckie, R.D.: A meshless method to simulate solute transport in heterogeneous porous media. Adv. Water Resour. 32(3), 413–429 (2009). doi: 10.1016/j.advwatres.2008.12.005 CrossRefGoogle Scholar
  94. Herrera, P.A., Valocchi, A.J., Beckie, R.D.: A multidimensional streamline-based method to simulate reactive solute transport in heterogeneous porous media. Adv. Water Resour. 33(7), 711–727 (2010). doi: 10.1016/j.advwatres.2010.03.001 CrossRefGoogle Scholar
  95. Holzner, M., Morales, V.L., Willmann, M., Dentz, M.: Intermittent lagrangian velocities and accelerations in three-dimensional porous medium flow. Phys. Rev. E (2015). doi: 10.1103/PhysRevE.92.013015 Google Scholar
  96. Hoteit, H., Erhel, J., Mos, R., Philippe, B., Ackerer, P.: Numerical reliability for mixed methods applied to flow problems in porous media. Comput. Geosci. 6(2), 161–194 (2002a). doi: 10.1023/A:1019988901420 CrossRefGoogle Scholar
  97. Hoteit, H., Mose, R., Younes, A., Lehmann, F., Ackerer, P.: Three-dimensional modeling of mass transfer in porous media using the mixed hybrid finite elements and the random-walk methods. Math. Geol. 34(4), 435–456 (2002b). doi: 10.1023/A:1015083111971 CrossRefGoogle Scholar
  98. Hu, L.Y.: Gradual deformation and iterative calibration of Gaussian-related stochastic models. Math. Geol. 32(1), 87–108 (2000)CrossRefGoogle Scholar
  99. Jimenez-Hornero, F., Giraldez, J., Laguna, A., Pachepsky, Y.: Continuous time random walks for analyzing the transport of a passive tracer in a single fissure. Water Resour. Res. (2005). doi: 10.1029/2004WR003852 Google Scholar
  100. Kang, P.K., Dentz, M., Le Borgne, T., Juanes, R.: Spatial Markov model of anomalous transport through random lattice networks. Phys. Rev. Lett. (2011). doi: 10.1103/PhysRevLett.107.180602
  101. Kang, P.K., de Anna, P., Nunes, J.P., Bijeljic, B., Blunt, M., Juanes, R.: Pore-scale intermittent velocity structure underpinning anomalous transport through 3D porous media. Geophys. Res. Lett. 41(17), 6184–6190 (2014). doi: 10.1002/2014GL061475 CrossRefGoogle Scholar
  102. Kang, P.K., Le Borgne, T., Dentz, T., Bour, O., Juanes, R.: Impact of velocity correlation and distribution on transport in fractured media: field evidence and theoretical model. Water Resour. Res. 51(2), 940–959 (2015). doi: 10.1002/2014WR015799 CrossRefGoogle Scholar
  103. Kenkre, V.M., Montroll, E.W., Shlesinger, M.F.: Generalized master equations for continuous-time random walks. J. Stat. Phys. 9(1), 45–50 (1973)CrossRefGoogle Scholar
  104. Khvoenkova, N., Delorme, M.: An optimal method to model transient flows in 3D discrete fracture network. IAMG Conf. 2011, 1238–1249 (2011). doi: 10.5242/iamg.2011.0088 Google Scholar
  105. Kim, I.C., Torquato, S.: Effective conductivity of suspensions of overlapping spheres. J. Appl. Phys. 71(6), 2727–2735 (1992). doi: 10.1063/1.351046 CrossRefGoogle Scholar
  106. Kinzelbach, W.: The random walk method in pollutant transport simulation. In: Groundwater flow and quality modelling, Springer, Berlin, pp 227–245 (1988)Google Scholar
  107. Kinzelbach, W., Uffink, G.: The random walk method and extensions in groundwater modelling. Processes in Porous Media, vol. Transport. Springer, Netherlands (1991)Google Scholar
  108. Kitanidis, P.: The concept of the dilution index. Water Resour. Res. 30(7), 2011–2026 (1994). doi: 10.1029/94WR00762 CrossRefGoogle Scholar
  109. Klafter, J., Silbey, R.: Derivation of the continuous-time random-walk equation. Phys. Rev. Lett. 44(2), 55–58 (1980). doi: 10.1103/PhysRevLett.44.55
  110. Klafter, J., Sokolov, I.: Anomalous diffusion spreads its wings. Phys. World 18(8), 29–32 (2005)CrossRefGoogle Scholar
  111. Koplik, J., Redner, S., Wilkinson, D.: Transport and dispersion in random networks with percolation disorder. Phys. Rev. A 37(7), 2619–2636 (1988). doi: 10.1103/PhysRevA.37.2619 CrossRefGoogle Scholar
  112. Kosakowski, G.: Anomalous transport of colloids and solutes in a shear zone. J. Contam. Hydrol. 72(1–4), 23–46 (2004). doi: 10.1016/j.jconhyd.2003.10.005 CrossRefGoogle Scholar
  113. Kosakowski, G., Berkowitz, B.: Flow pattern variability in natural fracture intersections. Geophys. Res. Lett. 26(12), 1765–1768 (1999). doi: 10.1029/1999GL900344 CrossRefGoogle Scholar
  114. Kosakowski, G., Berkowitz, B., Scher, H.: Analysis of field observations of tracer transport in a fractured till. J. Contam. Hydrol. 47(1), 29–51 (2001). doi: 10.1016/S0169-7722(00)00140-6 CrossRefGoogle Scholar
  115. LaBolle, E.M., Quastel, J., Fogg, G.E.: Diffusion theory for transport in porous media: transition-probability densities of diffusion processes corresponding to advection-dispersion equations. Water Resour. Res. 34(7), 1685–1693 (1998)CrossRefGoogle Scholar
  116. Landereau, P., Noetinger, B., Quintard, M.: Quasi-steady two-equation models for diffusive transport in fractured porous media: large-scale properties for densely fractured systems. Adv. Water Resour. 24(8), 863–876 (2001). doi: 10.1016/S0309-1708(01)00015-X CrossRefGoogle Scholar
  117. Le Borgne, T., Gouze, P.: Non-fickian dispersion in porous media: 2. Model validation from measurements at different scales. Water Resour. Res. (2008). doi: 10.1029/2007WR006279 Google Scholar
  118. Le Borgne, T., Dentz, M., Carrera, J.: A Lagrangian statistical model for transport in highly heterogeneous velocity fields. Phys. Rev. Lett. (2008a). doi: 10.1103/PhysRevLett.101.090601
  119. Le Borgne, T., Dentz, M., Carrera, J.: Spatial Markov processes for modeling lagrangian particle dynamics in heterogeneous porous media. Phys. Rev. E (2008b). doi: 10.1103/PhysRevE.78.026308 Google Scholar
  120. Le Borgne, T., Dentz, M., Bolster, D., Carrera, J., de Dreuzy, J.R., Davy, P.: Non-fickian mixing: Temporal evolution of the scalar dissipation rate in heterogeneous porous media. Adv. Water Resour. 33(12), 1468–1475 (2010). doi: 10.1016/j.advwatres.2010.08.006 CrossRefGoogle Scholar
  121. Le Borgne, T., Dentz, M., Davy, P., Bolster, D., Carrera, J., de Dreuzy, J.R., Bour, O.: Persistence of incomplete mixing: a key to anomalous transport. Phys. Rev. E (2011). doi: 10.1103/PhysRevE.84.015301 Google Scholar
  122. Le Borgne, T., Dentz, M., Villermaux, E.: Stretching, coalescence and mixing in porous media. Phys. Rev. Lett. (2013). doi: 10.1103/PhysRevLett.110.204501
  123. Le Borgne, T., Dentz, M., Villermaux, E.: The lamellar description of mixing in porous media. J. Fluid Mech. 770, 458–498 (2015). doi: 10.1017/jfm.2015.117 CrossRefGoogle Scholar
  124. Lejay, A., Pichot, G.: Simulating diffusion processes in discontinuous media: benchmark tests. J. Comput. Phys. 314, 384–413 (2016)CrossRefGoogle Scholar
  125. Leray, S., de Dreuzy, J.R., Aquilina, L., Vergnaud-Ayraud, V., Labasque, T., Bour, O., Le Borgne, T.: Temporal evolution of age data under transient pumping conditions. J. Hydrol. 511, 555–566 (2014). doi: 10.1016/j.jhydrol.2014.01.064 CrossRefGoogle Scholar
  126. Liu, H., Zhang, Y., Zhou, Q., Molz, F.: An interpretation of potential scale dependence of the effective matrix diffusion coefficient. J. Contam. Hydrol. 90(1–2), 41–57 (2007). doi: 10.1016/j.jconhyd.2006.09.006 CrossRefGoogle Scholar
  127. Liu, H.H., Bodvarsson, G.S., Pan, L.: Determination of particle transfer in random walk particle methods for fractured porous media. Water Resour. Res. 36(3), 707–713 (2000). doi: 10.1029/1999WR900323 CrossRefGoogle Scholar
  128. Maier, U., Bürger, C.M.: An accurate method for transient particle tracking. Water Resour. Res. 49(5), 3059–3063 (2013)CrossRefGoogle Scholar
  129. Matheron, G., de Marsily, G.: Is transport in porous media always diffusive? A counterexample. Water Resour. Res. 16(5), 901–917 (1980). doi: 10.1029/WR016i005p00901 CrossRefGoogle Scholar
  130. McCarthy, J.F.: Effective permeability of sandstone-shale reservoirs by a random walk method. J. Phys. A Math. General 23(9), L445 (1990)CrossRefGoogle Scholar
  131. McCarthy, J.F.: Analytical models of the effective permeability of sand-shale reservoirs. Geophys. J. Int. 105(2), 513–527 (1991). doi: 10.1111/j.1365-246X.1991.tb06730.x CrossRefGoogle Scholar
  132. McCarthy, J.F.: Continuous-time random walks on random media. J. Phys. A Math. Gen. 26(11), 2495–2503 (1993a). doi: 10.1088/0305-4470/26/11/004 CrossRefGoogle Scholar
  133. McCarthy, J.F.:P Reservoir characterization : efficient random-walk methods for upscaling and image selection. In: SPE Asia Pacific Oil and Gas Conference, 8–10 February, Singapore 25334 (1993b)Google Scholar
  134. Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339(1), 1–77 (2000). doi: 10.1016/S0370-1573(00)00070-3 CrossRefGoogle Scholar
  135. Metzler, R., Glockle, W.G., Nonnenmacher, T.F.: Fractional model equation for anomalous diffusion. Phys. A Stat. Mech. Appl. 211(1), 13–24 (1994). doi: 10.1016/0378-4371(94)90064-7 CrossRefGoogle Scholar
  136. Metzler, R., Jeon, J.H., Cherstvy, A.G., Barkai, E.: Anomalous diffusion models and their properties: non-stationarity, non-ergodicity, and ageing at the centenary of single particle tracking. Phys. Chem. Chem. Phys. 16(44), 24128–24164 (2014). doi: 10.1039/c4cp03465a CrossRefGoogle Scholar
  137. Michalak, A.M., Kitanidis, P.K.: Macroscopic behavior and random-walk particle tracking of kinetically sorbing solutes. Water Resour. Res. 36(8), 2133–2146 (2000). doi: 10.1029/2000WR900109 CrossRefGoogle Scholar
  138. Monaghan, J.J.: Smoothed particle hydrodynamics. Rep. Prog. Phys. 68(8), 1703–1759 (2005). doi: 10.1088/0034-4885/68/8/R01 CrossRefGoogle Scholar
  139. Mosé, R., Siegel, P., Ackerer, P., Chavent, G.: Application of the mixed hybrid finite element approximation in a groundwater model: Luxury or necessity? Water Resour. Res. 30(11), 3001–3012 (1994). doi: 10.1029/94WR01786 CrossRefGoogle Scholar
  140. Narasimhan, T.N., Pruess, K.z: MINC: An approach for analyzing transport in strongly heterogeneous systems. In: Flow, G., Modeling, Q. (eds.) Springer Netherlands, 224, pp 375–391 (1988)Google Scholar
  141. Néel, M.C., Rakotonasyl, S.H., Bauer, D., Joelson, M., Fleury, M.: All order moments and other functionals of the increments of some non-markovian processes. J. Stat. Mech. Theory Experiment (2011). doi: 10.1088/1742-5468/2011/02/P02006 Google Scholar
  142. Néel, M.C., Bauer, D., Fleury, M.: Model to interpret pulsed-field-gradient NMR data including memory and superdispersion effects. Phys. Rev. E (2014). doi: 10.1103/PhysRevE.89.062121 Google Scholar
  143. Noetinger, B.: An explicit formula for computing the sensitivity of the effective conductivity of heterogeneous composite materials to local inclusion transport properties and geometry. SIAM Multiscale Model. Simul. 11(3), 907–924 (2013). doi: 10.1137/120884961 CrossRefGoogle Scholar
  144. Noetinger, B.: A quasi steady state method for solving transient Darcy flow in complex 3D fractured networks accounting for matrix to fracture flow. J. Comput. Phys. 283, 205–223 (2015). doi: 10.1016/ CrossRefGoogle Scholar
  145. Noetinger, B., Estebenet, T.: Up-scaling of double porosity fractured media using continuous-time random walks methods. Transp. Porous Media 39(3), 315–337 (2000). doi: 10.1023/A:1006639025910 CrossRefGoogle Scholar
  146. Noetinger, B., Gautier, Y.: Use of the Fourier-Laplace transform and of diagrammatical methods to interpret pumping tests in heterogeneous reservoirs. Adv. Water Resour. 21(7), 581–590 (1998)CrossRefGoogle Scholar
  147. Noetinger, B., Jarrige, N.: A quasi steady state method for solving transient Darcy flow in complex 3D fractured networks. J. Comput. Phys. 231(1), 23–38 (2012). doi: 10.1016/ CrossRefGoogle Scholar
  148. Noetinger, B., Estebenet, T., Landereau, P.: A direct determination of the transient exchange term of fractured media using a continuous time random walk method. Transp. Porous Media 44(3), 539–557 (2001a). doi: 10.1023/A:1010647108341 CrossRefGoogle Scholar
  149. Noetinger, B., Estebenet, T., Quintard, M.: Up scaling of fractured media: Equivalence between the large scale averaging theory and the continuous time random walk method. Transp. Porous Media 43(3), 581–596 (2001b). doi: 10.1023/A:1010733724498 CrossRefGoogle Scholar
  150. Nœtinger, B., Artus, V., Ricard, L.: Dynamics of the water-oil front for two-phase, immiscible flow in heterogeneous porous media. 2-Isotropic media. Transp. Porous Media 56(3), 305–328 (2004). doi: 10.1023/ CrossRefGoogle Scholar
  151. Nunes, J.P., Bijeljic, B., Blunt, M.J.: Time-of-flight distributions and breakthrough curves in heterogeneous porous media using a pore-scale streamline tracing algorithm. Transp. Porous Media 109(2), 317–336 (2015). doi: 10.1007/s11242-015-0520-y CrossRefGoogle Scholar
  152. O’Brien, G.S., Bean, C.J., McDermott, F.: Numerical investigations of passive and reactive flow through generic single fractures with heterogeneous permeability. Earth Planet. Sci. Lett. 213(3–4), 271–284 (2003a). doi: 10.1016/S0012-821X(03)00342-X CrossRefGoogle Scholar
  153. O’Brien, G.S., Bean, C.J., McDermott, F.: A numerical study of passive transport through fault zones. Earth Planet. Sci. Lett. 214(3–4), 633–643 (2003b). doi: 10.1016/S0012-821X(03)00398-4 CrossRefGoogle Scholar
  154. Odeh, A.S.: Unsteady-state behavior of naturally fractured reservoirs. SPE J. 5(1), 60–66 (1965). doi: 10.2118/966-PA CrossRefGoogle Scholar
  155. Oliver, D.S., Cunha, L.B., Reynolds, A.C.: Markov chain Monte Carlo methods for conditioning a permeability field to pressure data. Math. Geol. 29(1), 61–91 (1997)CrossRefGoogle Scholar
  156. O’Shaughnessy, B., Procaccia, I.: Diffusion on fractals. Phys. Rev. A 32(5), 3073–3083 (1985). doi: 10.1103/PhysRevA.32.3073 CrossRefGoogle Scholar
  157. Ottino, J.M.: The kinematics of mixing: stretching, chaos and transport. Cambridge University Press, Cambridge (1989)Google Scholar
  158. Painter, S., Cvetkovic, V.: Upscaling discrete fracture network simulations: an alternative to continuum transport models. Water Resour. Res. (2005). doi: 10.1029/2004WR003682 Google Scholar
  159. Painter, S., Cvetkovic, V., Mancillas, J., Pensado, O.: Time domain particle tracking methods for simulating transport with retention and first-order transformation. Water Resour. Res. (2008). doi: 10.1029/2007WR005944 Google Scholar
  160. Pan, L., Bodvarsson, G.S.: Modeling transport in fractured porous media with the random-walk particle method: the transient activity range and the particle transfer probability. Water Resour. Res. (2002). doi: 10.1029/2001WR000901 Google Scholar
  161. Park, Y., de Dreuzy, J.R., Lee, K.K., Berkowitz, B.: Transport and intersection mixing in random fracture networks with power law length distributions. Water Resour. Res. 37(10), 2493–2501 (2001). doi: 10.1029/2000WR000131 CrossRefGoogle Scholar
  162. Park, Y., Lee, K., Kosakowski, G., Berkowitz, B.: Transport behavior in three-dimensional fracture intersections. Water Resour. Res. (2003). doi: 10.1029/2002WR001801 Google Scholar
  163. Pichot, G., Erhel, J., de Dreuzy, J.R.: A mixed hybrid mortar method for solving flow in discrete fracture networks. Appl. Anal. 89(10), 1629–1643 (2010). doi: 10.1080/00036811.2010.495333 CrossRefGoogle Scholar
  164. Pollock, D.W.: Semianalytical computation of path lines for finite-difference models. Ground Water 26(6), 743–750 (1988). doi: 10.1111/j.1745-6584.1988.tb00425.x CrossRefGoogle Scholar
  165. Qu, Z.X., Liu, Z.F., Wang, X.H., Zhao, P.: Finite analytic numerical method for solving two-dimensional quasi-Laplace equation. Numer. Methods Partial Differ. Equ. 30(6), 1755–1769 (2014). doi: 10.1002/num.21863 CrossRefGoogle Scholar
  166. Quintard, M., Whitaker, S.: One- and two-equation models for transient diffusion processes in two-phase systems. Adv. Heat Transf. 23, 369–464 (1993). doi: 10.1016/S0065-2717(08)70009-1 CrossRefGoogle Scholar
  167. Redner, S.: Transport due to random velocity fields. Phys. D 38(1–3), 287–290 (1989). doi: 10.1016/0167-2789(89)90207-8 CrossRefGoogle Scholar
  168. Risken, H.: The Fokker-Planck Equation. Springer, Heidelberg New York (1996)CrossRefGoogle Scholar
  169. Rivard, C., Delay, F.: Simulations of solute transport in fractured porous media using 2D percolation networks with uncorrelated hydraulic conductivity fields. Hydrogeol. J. 12(6), 613–627 (2004). doi: 10.1007/s10040-004-0363-z CrossRefGoogle Scholar
  170. Roberts, J.E., Thomas, J.M.: Mixed and hybrid methods. In: Handbook of Numerical Analysis 2, Finite Element Methods -part 1, Elsevier Science Publishers B.V. (North-Holland), pp 523–639 (1991)Google Scholar
  171. Robinet, J.C., Sardini, P., Delay, F., Hellmuth, K.H.: The effect of rock matrix heterogeneities near fracture walls on the residence time distribution (RTD) of solutes. Transp. Porous Media 72(3), 393–408 (2007). doi: 10.1007/s11242-007-9159-7 CrossRefGoogle Scholar
  172. Romary, T.: Integrating production data under uncertainty by parallel interacting markov chains on a reduced dimensional space. Comput. Geosci. 13(1), 103–122 (2009)CrossRefGoogle Scholar
  173. Romeu, R.K., Noetinger, B.: Calculation of internodal transmissivities in finite difference models of flow in heterogeneous porous media. Water Resour. Res. 31(4), 943–959 (1995). doi: 10.1029/94WR02422 CrossRefGoogle Scholar
  174. Roubinet, D., Irving, J.: Discrete-dual-porosity model for electric current flow in fractured rock. J. Geophys. Res. Solid Earth 119(2), 767–786 (2014). doi: 10.1002/2013JB010668 CrossRefGoogle Scholar
  175. Roubinet, D., Liu, H.H., de Dreuzy, J.R.: A new particle-tracking approach to simulating transport in heterogeneous fractured porous media. Water Resour. Res. (2010). doi: 10.1029/2010WR009371 Google Scholar
  176. Roubinet, D., de Dreuzy, J.R., Tartakovsky, D.M.: Particle-tracking simulations of anomalous transport in hierarchically fractured rocks. Comput. Geosci. 50(SI), 52–58 (2013). doi: 10.1016/j.cageo.2012.07.032 CrossRefGoogle Scholar
  177. Russian, A., Dentz, M., Gouze, P.: Time domain random walks for hydrodynamic transport in heterogeneous media. Water Resour. Res. (2016). doi: 10.1002/2015WR018511
  178. Saffman, P.G., Taylor, G.: The penetration of a fluid into a porous medium or Hele-Shaw cell containing a more viscous liquid. Proc. R. Soc. Lond. A Math. Phys. Eng. Sci. R. Soc. 245, 312–329 (1958)CrossRefGoogle Scholar
  179. Sahimi, M.: Flow and transport in porous media and fractured rock: from classical methods to modern approaches. Wiley, New York (2011)CrossRefGoogle Scholar
  180. Salamon, P., Fernandez-Garcia, D., Gomez-Hernandez, J.J.: A review and numerical assessment of the random walk particle tracking method. J. Contam. Hydrol. 87(3–4), 277–305 (2006). doi: 10.1016/j.jconhyd.2006.05.005 CrossRefGoogle Scholar
  181. Salles, J., Thovert, J.F., Delannay, R., Prevors, L., Auriault, J.L., Adler, P.: Taylor dispersion in porous media. Determination of the dispersion tensor. Phys. Fluids A 5(10), 2348–2376 (1993)CrossRefGoogle Scholar
  182. Scher, H., Lax, M.: Stochastic transport in a disordered solid I. Theory. Phys. Rev. B 7(10), 4491–4502 (1973a). doi: 10.1103/PhysRevB.7.4491 CrossRefGoogle Scholar
  183. Scher, H., Lax, M.: Stochastic transport in a disordered solid II. Impurity conduction. Phys. Rev. B 7(10), 4502–4519 (1973b). doi: 10.1103/PhysRevB.7.4502 CrossRefGoogle Scholar
  184. Scher, H., Margolin, G., Berkowitz, B.: Towards a unified framework for anomalous transport in heterogeneous media. Chem. Phys. 284(1–2), 349–359 (2002a). doi: 10.1016/S0301-0104(02)00558-X CrossRefGoogle Scholar
  185. Scher, H., Margolin, G., Metzler, R., Klafter, J., Berkowitz, B.: The dynamical foundation of fractal stream chemistry: the origin of extremely long retention times. Geophys. Res. Lett. (2002b). doi: 10.1029/2001GL014123 Google Scholar
  186. Semra, K., Ackerer, P., Mosé, R.: Three dimensional groundwater quality modeling in heterogeneous media. Water pollution II: Modeling, measuring and prediction, pp. 3–11. Computational Mechanics Publications, Southampton, UK (1993)Google Scholar
  187. Sen, P.: Time-dependent diffusion coefficient as a probe of permeability of the pore-wall. J. Chem. Phys. 119(18), 9871–9876 (2003). doi: 10.1063/1.1611477 CrossRefGoogle Scholar
  188. Sen, P.: Time-dependent diffusion coefficient as a probe of geometry. Concepts Magn. Resonance Part A 23A(1), 1–21 (2004). doi: 10.1002/cmr.a.20017 CrossRefGoogle Scholar
  189. Sen, P., Schwartz, L., Mitra, P., Halperin, B.: Surface relaxation and the long-time diffusion coefficient in porous media: periodic geometries. Phys. Rev. B 49(1), 215–225 (1994). doi: 10.1103/PhysRevB.49.215 CrossRefGoogle Scholar
  190. Srinivasan, G., Tartakosky, D.M., Dentz, M., Viswanathan, H., Berkowitz, B., Robinson, B.A.: Random walk particle tracking simulations of non-fickian transport in heterogeneous media. J. Comput. Phys. 229(11), 4304–4314 (2010). doi: 10.1016/ CrossRefGoogle Scholar
  191. Sun, N.Z.: A finite cell method for simulating the mass transport process in porous media. Water Resour. Res. 35(12), 3649–3662 (1999). doi: 10.1029/1999WR900187 CrossRefGoogle Scholar
  192. Sun, N.Z.: Modeling biodegradation processes in porous media by the finite cell method. Water Resour. Res. 38, 3 (2002). doi: 10.1029/2000WR000198 Google Scholar
  193. Tallakstad, K.T., Knudsen, H.A., Ramstad, T., Lovoll, G., Maloy, K.J., Toussaint, R., Flekkoy, E.G.: Steady-state two-phase flow in porous media: statistics and transport properties. Phys. Rev. Lett. 102(7), (2009). doi: 10.1103/PhysRevLett.102.074502
  194. Tang, C.: Diffusion-limited aggregation and the Saffman-Taylor problem. Phys. Rev. A 31(3), 1977–1979 (1985). doi: 10.1103/PhysRevA.31.1977 CrossRefGoogle Scholar
  195. Tang, D.H., Frind, E.O., Sudicky, E.A.: Contaminant transport in fractured porous media: Analytical solution for a single fracture. Water Resour. Res. 17(3), 555–564 (1981). doi: 10.1029/WR017i003p00555 CrossRefGoogle Scholar
  196. Tartakovsky, A.M., Meakin, P.: Pore scale modeling of immiscible and miscible fluid flows using smoothed particle hydrodynamics. Adv. Water Resour. 29(10), 1464–1478 (2006). doi: 10.1016/j.advwatres.2005.11.014 CrossRefGoogle Scholar
  197. Taylor, G.I.: Diffusion and mass transport in tubes. Proc. Phys. Soc. Sect. B 67(420), 857–869 (1954). doi: 10.1088/0370-1301/67/12/301 CrossRefGoogle Scholar
  198. Teodorovich, E., Spesivtsev, P., Nœtinger, B.: A stochastic approach to the two-phase displacement problem in heterogeneous porous media. Transp. Porous Media 87(1), 151–177 (2011). doi: 10.1007/s11242-010-9673-x CrossRefGoogle Scholar
  199. Tompson, A.F.B., Gelhar, L.W.: Numerical simulation of solute transport in three-dimensional, randomly heterogeneous porous media. Water Resour. Res. 26(10), 2541–2562 (1990). doi: 10.1029/WR026i010p02541 CrossRefGoogle Scholar
  200. Uffink, G.J.M.: A random walk method for the simulation of macrodispersion in a stratified aquifer. In: 18th General Assembly Proceedings Symposium, IAHS Publications. 146, IAHS, Wallingford, UK., Hamburg, Germany (1985)Google Scholar
  201. Wang, Y.F., Liu, Z.F., Wang, X.H.: Finite analytic numerical method for three-dimensional fluid flow in heterogeneous porous media. J. Comput. Phys. 278, 169–181 (2014). doi: 10.1016/ CrossRefGoogle Scholar
  202. Warren, J.E., Root, P.J.: The behavior of naturally fractured reservoirs. Soc. Petrol. Eng. J. 3(3), 245–255 (1963)CrossRefGoogle Scholar
  203. Wen, X.H., Gomez-Hernandez, J.J.: The constant displacement scheme for tracking particles in heterogeneous aquifers. Ground Water 34(1), 135–142 (1996). doi: 10.1111/j.1745-6584.1996.tb01873.x CrossRefGoogle Scholar
  204. Willmann, M., Carrera, J., Snchez-Vila, X.: Transport upscaling in heterogeneous aquifers: What physical parameters control memory functions? Water Resour. Res. 44, 12 (2008). doi: 10.1029/2007WR006531 CrossRefGoogle Scholar
  205. Witten, T.A., Sander, L.M.: Diffusion-limited aggregation. Phys. Rev. B 27(9), 5686–5697 (1983). doi: 10.1103/PhysRevB.27.5686 CrossRefGoogle Scholar
  206. Zheng, C., Bennett, G.D.: Applied contaminant transport modeling, 2nd edn, p. 440. Wiley, Hoboken (2002)Google Scholar
  207. Zimmermann, S., Koumoutsakos, P., Kinzelbach, W.: Simulation of pollutant transport using a particle method. J. Comput. Phys. 173(1), 322–347 (2001). doi: 10.1006/jcph.2001.6879 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  • Benoit Noetinger
    • 1
  • Delphine Roubinet
    • 2
  • Anna Russian
    • 3
  • Tanguy Le Borgne
    • 4
  • Frederick Delay
    • 5
  • Marco Dentz
    • 6
  • Jean-Raynald de Dreuzy
    • 7
  • Philippe Gouze
    • 3
  1. 1.IFPENRueil-MalmaisonFrance
  2. 2.Applied and Environmental Geophysics Group, Institute of Earth SciencesUniversity of LausanneLausanneSwitzerland
  3. 3.Géosciences, CNRSUniversité de MontpellierMontpellierFrance
  4. 4.CNRS, Géosciences Rennes, UMR 6118Université de Rennes 1RennesFrance
  5. 5.Laboratoire dHydrologie et de Géochimie de Strasbourg, CNRS UMR 7517Univ. Strasbourg/EOSTStrasbourgFrance
  6. 6.Institute of Environmental Assessment and Water ResearchSpanish National Research CouncilBarcelonaSpain
  7. 7.CNRS, UMR 6118Géosciences RennesRennesFrance

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