Diffusion and trapping in heterogeneous media: An inhomogeneous continuous time random walk approach

Advances in Water Resources

Volumn 49, Issue , 2012, Pages 13-22

  Dentz, Marco ^a   Gouze, Philippe ^b   Russian, Anna ^a   Dweik, Jalal ^b   Delay, Frederick ^c  

^a INSTITUTE OF ENVIRONMENTAL ASSESSMENT AND WATER RESEARCH   (Spain)

^b UNIV MONTPELLIER   (France)

^c UNIVERSITÉ DE STRASBOURG   (France)

Author keywords

Continuous time random walks; Diffusion in random media; Multirate mass transfer; Stochastic modeling; Time domain random walks

Indexed keywords

CONTINUOUS TIME RANDOM WALK APPROACHES; CONTINUOUS TIME RANDOM WALKS; CONTINUOUS-TIME RANDOM WALK; DIFFUSION BEHAVIOR; DIFFUSION EQUATIONS; DIFFUSION PROPERTIES; HETEROGENEOUS MEDIA; HETEROGENEOUS MEDIUM; MULTI RATE; PARTICLE FORMULATION; POWER-LAW; RANDOM MEDIA; RANDOM WALK; SPATIALLY INHOMOGENEOUS; TIME-DOMAIN RANDOM WALKS; TRANSFER PROPERTIES; TRANSITION TIME; TRAPPING TIME;

LAPLACE TRANSFORMS; PARTIAL DIFFERENTIAL EQUATIONS; RANDOM PROCESSES; STOCHASTIC MODELS;

DIFFUSION;

HETEROGENEOUS MEDIUM; MASS TRANSFER; NUMERICAL MODEL; RANDOM WALK METHOD; SPATIAL ANALYSIS; STOCHASTICITY;

EID: 84866542920     PISSN: 03091708     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.advwatres.2012.07.015     Document Type: Article

References (43)

1. Haus J.W., Kehr K.W. Diffusion in regular and disordered lattices. Phys Rep 1987, 150:263-406.
2. Havlin S., Ben-Avraham D. Diffusion in disordered media. Adv Phys 2002, 51:187-292.
3. Dean D.S., Drummond I.T., Horgan R.R. Effective transport properties for diffusion in random media. J Stat Mech 2007, P07013.
4. Delay F., Porel G., Sardini P. Modelling diffusion in a heterogeneous rock matrix with a time-domain Lagrangian method and an inversion procedure. CR Geosci 2002, 334:967-973.
5. Gouze P., Melean Z., Le Borgne T., Dentz M., Carrera J. Non-fickian dispersion in porous media explained by heterogeneous microscale matrix diffusion. Water Resour Res 2008, 44:W11416.
6. King P.R. The use of field theoretic methods for the study of flow in a heterogeneous porous medium. J Phys A: Math Gen 1987, 20:3935-3947.
7. Drummond I.T., Horgan R.R. The effective permeability of a random medium. J Phys A: Math Gen 1987, 20:4661-4672.
8. McCarthy J.F. Continuous-time random walks on random media. J Phys A: Math Gen 1993, 26:2495-2503.
9. Codling E.A. Random walk models in biology. JR Soc Interface 2008, 5:813-834.
10. Pabitra S. Time-dependent diffusion coefficient as a probe of geometry. Concepts Magn Reson Part A 2004, 23A:1-21.
11. Patankar S.H. Numerical heat transfer and fluid flow 1980, Taylor and Francis.
12. Crank J. The mathematics of diffusion 1975, Oxford University Press.
13. Kinzelbach W. The random walk method in pollutant transport simulation. NATO ASI Series C 1987, vol. 224:227-246. E. Custodio (Ed.).
14. Risken H. The Fokker-Planck equation 1996, Springer, Heidelberg, New York.
15. James S.C., Chrysikopoulos C.V. An efficient particle tracking equation with a specified spatial step for the solution of the diffusion equation. Chem Eng Sci 2001, 56:6535-6543.
16. Delay F., Ackerer P., Danquigny C. Simulating solute transport in porous or fractured formations using random walk particle tracking: A review. Vadose Zone J 2005, 4:360-379.
17. Banton O., Delay F., Porel G. A new time domain random walk method for solute transport in 1-D heterogeneous media. Ground Water 1997, 35:1008-1013.
18. Noetinger B., Estebenet T. Up-scaling of double porosity fractured media using continuous-time random walks methods. Transp Porous Media 2000, 39:315-337.
19. Reimus P.W., James S.C. Determining the random time step in a constant spatial step particle tracking algorithm. Chem Eng Sci 2002, 57:4429-4434.
20. Delay F., Bodin J. Time domain random walk method to simulate transport by advection-diffusion and matrix diffusion in fracture networks. Geophys Res Lett 2001, 28:4051-4054.
21. 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 2003, 208:297-304.
22. Montroll E.W., Weiss G.H. Random walks on lattices, 2. J Math Phys 1965, 6(2):167.
23. Scher H., Lax M. Stochastic transport in a disordered solid. I. Theory. Phys Rev B 1973, 7(1):4491-4502.
24. Zumofen G., Klafter J. Scale-invariant motion in intermittent chaotic systems. Phys Rev E 1993, 47:851-863.
25. Berkowitz B., Cortis A., Dentz M., Scher H. Modeling non-fickian transport in geological formations as a continuous time random walk. Rev Geophys 2006, 44:RG2003.
26. 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 2002, 38(10):1191. 10.1029/2001WR001030.
27. Goychuk I. Quantum dynamics with non-Markovian fluctuating parameters. Phys Rev E 2004, 70:016109.
28. Chechkin A.V., Gorenflo R., Sokolov I.M. Fractional diffusion in inhomogeneous media. J Phys A 2005, 38:L679-L684.
29. Kenkre V.M., Montroll E.W., Shlesinger M.F. Generalized master equations for continuous-time random walks. J Stat Phys 1973, 9(1):45-50.
30. Abramowitz M., Stegun I.A. Handbook of mathematical functions 1972, Dover Publications, New York.
31. Bouchaud J.P., Georges A. Anomalous diffusion in disordered media: statistical mechanisms, models and physical applications. Phys Rep 1990, 195(4,5):127-293.
32. Haggerty R., Gorelick S.M. Multiple-rate mass transfer for modeling diffusion and surface reactions in media with pore-scale heterogeneity. Water Resour Res 1995, 31(10):2383-2400.
33. Carrera J., Sanchez-Vila X., Benet I., Medina A., Galarza G., Guimera J. On matrix diffusion, formulations, solutions methods and qualitative effects. Hydrogeol J 1998, 6(1):178-190.
34. Dentz M., Berkowitz B. Transport behavior of a passive solute in continuous time random walks and multirate mass transfer. Water Resour Res 2003, 39(5):1111.
35. Benson D.A., Meerschaert M.M. A simple and efficient random walk solution of multi-rate mobile/immobile mass transport equations. Adv Water Resour 2009, 32:532-539.
36. Margolin G., Dentz M., Berkowitz B. Continuous time random walk and multirate mass transfer modeling of sorption. Chem Phys 2003, 295:71-80.
37. Delay F., Kaczmaryk A., Ackerer P. Inversion of a Lagrangian time domain random walk (TDRW) approach to one-dimensional transport by derivation of the analytical sensitivities to parameters. Adv Water Resour 2008, 31:484-502.
38. Haggerty R., McKenna S.A., Meigs L.C. On the late time behavior of tracer test breakthrough curves. Water Resour Res 2000, 36(12):3467-3479.
39. Harvey C.F., Gorelick S.M. Temporal moment-generating equations: modeling transport and mass transfer in heterogeneous aquifers. Water Resour Res 1995, 31(8):1895-1911.
40. Matheron G. Composition des perméabilités en milieu poreux. Rev Inst Franc Petrole 1968, 23(2):201-218.
41. Carslaw H.S., Jaeger J.C. Conduction of heat in solids 2008, Oxford University Press.
42. Wittebroodt C., Savoye S., Gouze P. Influence of initial iodide concentration on the iodide uptake by the Argillite of Tournemire. Phys Chem Earth 2008, 33(14-16):943-948.
43. Haggerty R., Harvey C.F., Freiherr von Schwerin C., Meigs L.C. What controls the apparent timescale of solute mass transfer in aquifers and soils? A comparison of experimental results. Water Resour Res 2004, 40:W01510.