Continuous-time random-walk model of transport in variably saturated heterogeneous porous media

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 81, Issue 3, 2010, Pages

  Zoia, Andrea ^a   Néel, Marie Christine ^b   Cortis, Andrea ^c  

^a CEA SACLAY   (France)

^b UAPV   (France)

^c LAWRENCE BERKELEY NATIONAL LABORATORY   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

CONTINUOUS TIME; CONTINUOUS-TIME RANDOM WALK; ENSEMBLE AVERAGES; FLUID FLOW; HETEROGENEOUS POROUS MEDIA; MONTE CARLO SIMULATION; PARTICLE RETENTION; SOLUTE MIGRATION; VARIABLY SATURATED;

BOUNDARY CONDITIONS; COMPUTER SIMULATION; CONTINUOUS TIME SYSTEMS; MONTE CARLO METHODS; NONLINEAR EQUATIONS; POROUS MATERIALS;

FLOW OF FLUIDS;

EID: 77749344753     PISSN: 15393755     EISSN: 15502376     Source Type: Journal    
DOI: 10.1103/PhysRevE.81.031104     Document Type: Article

References (60)

1. M. Muskat, The Flow of Homogeneous Fluids Through Porous Media (McGraw Hill, New York, 1937).
2. R. Hilfer, Adv. Chem. Phys. 92, 299 (1996). 10.1002/9780470141519.ch3
3. J. Bear, Dynamics of Fluids in Porous Media (American Elsevier, New York, 1972).
4. M. Sahimi, Flow and Transport in Porous Media and Fractured Rock (VCH, Weinheim, 1995).
5. G. De Marsily, Quantitative Hydrogeology, Groundwater Hydrology for Engineers (Academic Press, New York, 1986).
6. A. Cortis and C. Knudby, Water Resour. Res. 42, W10201 (2006). 10.1029/2006WR005227
7. M. Bromly and C. Hinz, Water Resour. Res. 40, W07402 (2004). 10.1029/2003WR002579
8. M. Levy and B. Berkowitz, J. Contam. Hydrol. 64, 203 (2003). 10.1016/S0169-7722(02)00204-8
9. F. Delay, P. Ackerer and C. Danquigny, Vadose Zone J. 4, 360 (2005). 10.2136/vzj2004.0125
10. M. Bücker-Gittel, U. Mohrlok, and G. H. Jirka, IAHS-AISH Publication No. 277, 2003, pp. 17-21.
11. M. Magdziarz, A. Weron, and K. Weron, Phys. Rev. E 75, 016708 (2007). 10.1103/PhysRevE.75.016708
12. R. N. Bhattacharya, V. K. Gupta, and G. Sposito, Soil Sci. Soc. Am. J. 40, 465 (1976);
13. R. N. Bhattacharya, V. K. Gupta, and G. Sposito, Soil Sci. Soc. Am. J. 40, NP-a (1976).
14. R. Metzler and J. Klafter, Phys. Rep. 339, 1 (2000). 10.1016/S0370- 1573(00)00070-3
15. H. Scher and M. Lax, Phys. Rev. B 7, 4491 (1973). 10.1103/PhysRevB.7.4491
16. B. Berkowitz, A. Cortis, M. Dentz, and H. Scher, Rev. Geophys. 44, RG2003 (2006). 10.1029/2005RG000178
17. E. Gerolymatou, I. Vardoulakis, and R. Hilfer, J. Phys. D 39, 4104 (2006). 10.1088/0022-3727/39/18/022
18. L. A. Richards, Phys. 1, 318 (1931). 10.1063/1.1745010
19. Y. Pachepsky, D. Timlin, and W. Rawls, J. Hydrol. 272, 3 (2003). 10.1016/S0022-1694(02)00251-2
20. A. Cortis and B. Berkowitz, Soil Sci. Soc. Am. J. 68, 1539 (2004);
21. A. Cortis and B. Berkowitz, Soil Sci. Soc. Am. J. 69, 285 (2005).
22. C.-H. Park, C. Beyer, S. Bauer, and O. Kolditz, Geosci. J. 12, 285 (2008). 10.1007/s12303-008-0029-2
23. H. Risken, The Fokker-Planck Equation, 2nd ed. (Springer, Berlin, 1989).
24. E. M. LaBolle, J. Quastel, G. E. Fogg, and J. Gravner, Water Resour. Res. 36, 651 (2000). 10.1029/1999WR900224
25. E. M. LaBolle, J. Quastel, and G. E. Fogg, Water Resour. Res. 34, 1685 (1998). 10.1029/98WR00319
26. G. J. M. Uffink, Proceedings of the Hamburg Symposium, August, 1983 (IAHS, Institute of Hydrology, Wallingord, UK 1985), pp. 103-114.
27. R. H. Brooks and A. T. Corey, J. Irrig. Drain. 92, 61 (1966).
28. M. T. Van Genuchten, Soil Sci. Soc. Am. J. 44, 892 (1980).
29. A. Cortis, C. Gallo, H. Scher, and B. Berkowitz, Water Resour. Res. 40, W04209 (2004). 10.1029/2003WR002750
30. B. Berkowitz, A. Cortis, I. Dror, and H. Scher, Water Resour. Res. 45, W02201 (2009). 10.1029/2008WR007342
31. A. Cortis and A. Zoia, Phys. Rev. E 80, 011122 (2009). 10.1103/PhysRevE.80.011122
32. H. Scher, G. Margolin, and B. Berkowitz, Chem. Phys. 284, 349 (2002). 10.1016/S0301-0104(02)00558-X
33. R. Kimmich, Chem. Phys. 284, 253 (2002). 10.1016/S0301-0104(02)00552-9
34. S. C. Taylor, W. D. Hoff, M. A. Wilson, and K. M. Green, J. Mater. Sci. Lett. 18, 1925 (1999). 10.1023/A:1006677014070
35. M. Küntz and P. Lavallee, J. Phys. D 34, 2547 (2001). 10.1088/0022-3727/34/16/322
36. A. El-G. El Abd and J. J. Milczarek, J. Phys. D 37, 2305 (2004). 10.1088/0022-3727/37/16/013
37. Y. Pachepsky, D. Timlin, and W. Rawls, J. Hydrol. 279, 290 (2003). 10.1016/S0022-1694(03)00195-1
38. D. Or and T. A. Ghezzehei, Transp. Porous Media 68, 129 (2007). 10.1007/s11242-006-9060-9
39. R. Metzler, E. Barkai, and J. Klafter, Phys. Rev. Lett. 82, 3563 (1999). 10.1103/PhysRevLett.82.3563
40. R. Schumer, D. A. Benson, M. M. Meerschaert, and B. Bauemer, Water Resour. Res. 39, 1296 (2003). 10.1029/2003WR002141
41. M. C. Néel, A. Zoia, and M. Joelson, Phys. Rev. E 80, 056301 (2009). 10.1103/PhysRevE.80.056301
42. A. Cortis, Y. Chen, H. Scher, and B. Berkowitz, Phys. Rev. E 70, 041108 (2004). 10.1103/PhysRevE.70.041108
43. H. Ovdat and B. Berkowitz, Adv. Water Resour. 30, 2373 (2007). 10.1016/j.advwatres.2007.05.011
44. M. Han, S. Youssef, E. Rosenberg, M. Fleury, and P. Levitz, Phys. Rev. E 79, 031127 (2009). 10.1103/PhysRevE.79.031127
45. P. Beltrame, P. Hänggi, and U. Thiele, EPL 86, 24006 (2009). 10.1209/0295-5075/86/24006
46. B. Maryshev, M. Joelson, D. Lyubimov, T. Lyubimova, and M. C. Néel, J. Phys. A: Math. Theor. 42, 115001 (2009). 10.1088/1751-8113/42/ 11/115001
47. M. Th. Van Genuchten and P. J. Wierenga, Soil Sci. Soc. Am. J. 40, 473 (1976).
48. A. Zoia, C. Latrille, and A. Cartalade, Phys. Rev. E 79, 041125 (2009). 10.1103/PhysRevE.79.041125
49. - ε with ε > 0. This strong assumption implies that applying a time convolution with a bounded kernel will preserve the approximation. This was not necessary in, where only time-dependent velocity fields were addressed.
50. - α / Γ (1 - α) P 0.
51. R. Sánchez, B. A. Carreras, and B. Ph. van Milligen, Phys. Rev. E 71, 011111 (2005). 10.1103/PhysRevE.71.011111
52. B. Ph. van Milligen, R. Sánchez, and B. A. Carreras, Phys. Plasmas 11, 2272 (2004). 10.1063/1.1701893
53. J. F. Lutsko and J. P. Boon, Phys. Rev. E 77, 051103 (2008). 10.1103/PhysRevE.77.051103
54. B. Berkowitz, S. Emmanuel, and H. Scher, Water Resour. Res. 44, W03402 (2008). 10.1029/2007WR005906
55. A. Cortis, T. Harter, L. Hou, E. R. Atwill, A. Packman, and P. Green, Water Resour. Res. 42, W12S13 (2006). 10.1029/2006WR004897
56. A. Zoia, Phys. Rev. E 77, 041115 (2008). 10.1103/PhysRevE.77.041115
57. S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications (Gordon and Breach, New York, 1993).
58. B. Rubin, Fractional Integrals and Potentials (Longman, Harlow, 1996).
59. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, North-Holland Mathematical Studies 204 (Jan van Mill, Amsterdam, 2006).
60. F. Mainardi, in Fractals and Fractional Calculus in Continuum Mechanics, edited by, A. Carpinteri, and, F. Mainardi., CISM Courses and Lectures 378 (Springer, New York, 1997), pp. 291-348.