A simple and efficient random walk solution of multi-rate mobile/immobile mass transport equations

Advances in Water Resources

Volumn 32, Issue 4, 2009, Pages 532-539

  Benson, David A ^a   Meerschaert, Mark M ^b  

^a Department of Geology and Geological Engineering   (United States)

^b MICHIGAN STATE UNIVERSITY   (United States)

Author keywords

Mobile immobile; Multi rate mass transfer; Particle tracking; Subordination

Indexed keywords

MARKOV PROCESSES; PARTICLE BEAM TRACKING;

EXACT DISTRIBUTIONS; IMMOBILE PHASE; MASS TRANSPORT EQUATIONS; MEMORY FUNCTIONS; MOBILE STATE; MOBILE/IMMOBILE; MULTI-RATE MASS TRANSFER; PARTICLE TRACKING; PARTICLE-TRACKING METHODS; RANDOM WALKS; RATE EQUATIONS; RESIDENCE TIME; STOCHASTIC PROCESS; SUBORDINATION; TWO-STATE MARKOV CHAINS; WAITING-TIME;

MASS TRANSFER;

CHANNEL HYDRAULICS; COMPUTER SIMULATION; MARKOV CHAIN; MASS TRANSFER; RANDOM WALK METHOD; SEDIMENT TRANSPORT; STOCHASTICITY; TRACKING;

EID: 62349103035     PISSN: 03091708     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.advwatres.2009.01.002     Document Type: Article

References (38)

1. Bear J. Dynamics of fluids in porous media (1972), Elsevier, New York
2. Bochner S. Diffusion equations and stochastic processes. Proc Natl Acad Sci USA 85 (1949) 369-370
3. Brusseau M.L., Jessup R.E., and Rao P.S.C. Modeling the transport of solutes influenced by multiprocess nonequilibrium. Water Resour Res 25 9 (1989) 1971-1988
4. Carrera J., Sanchez-Vila X., Benet I., Medina A., Galarza G., and Guimera J. On matrix diffusion: formulations, solution methods and qualitative effects. Hydrogeol J 6 1 (1998) 178-190
5. Çinlar E. Introduction to stochastic processes (1975), Prentice-Hall, Englewood Cliffs, New Jersey
6. Coats K.H., and Smith B.D. Dead-end pore volume and dispersion in porous media. Soc Pet Eng J 4 (1964) 73-84
7. Delay F., and Bodin J. Time domain random walk method to simulate transport by advection-dispersion and matrix diffusion in fracture networks. Geophys Res Lett 28 21 (2001) 4051-4054
8. Einstein A. On the theory of the Brownian movement. Ann Phys 4 19 (1906) 371-381
9. Feller W. An introduction to probability theory and its applications. 2nd ed. Wiley series in probability and mathematical statistics vol. II (1971), Wiley, New York
10. Haggerty R., Wondzell S.M., and Johnson M.A. Power-law residence time distribution in the hyporheic zone of a 2nd-order mountain stream. Geophys Res Lett 29 13 (2002) 1640 10.1029/2002GL014743
11. Haggerty R., and Gorelick S.M. Multiple-rate mass transfer for modeling diffusion and surface reactions in media with pore-scale heterogeneity. Water Resour Res 31 10 (1995) 2383-2400
12. Haggerty R., McKenna S.A., and Meigs L.C. On the late-time behavior of tracer test breakthrough curves. Water Resour Res 36 12 (2000) 3467-3479
13. Hassan A.E., and Mohamed M.M. On using particle tracking methods to simulate transport in single-continuum and dual continua porous media. J Hydrol 275 (2002) 242-260
14. Kinzelbach W. The random walk method in pollutant transport simulation. In: Custodio E., et al. (Ed). Groundwater flow and quality modelling (1988), Reidel Publishing Company 227-245
15. LaBolle E.M., Fogg G.E., and Tompson A.F.B. Random-walk simulation of transport in heterogeneous porous media: local mass-conservation problem and implementation methods. Water Resour Res 32 3 (1996) 583-593
16. LaBolle E.M., Quastel J., Fogg G.E., and Gravner J. Diffusion processes in composite porous media and their numerical integration by random walks: generalized stochastic differential equations with discontinuous coefficients. Water Resour Res 36 3 (2000) 651-662
17. Liu H.H., Bodvarsson G.S., and Pan L. Determination of particle transfer in random walk particle methods for fractured porous media. Water Resour Res 36 3 (2000) 707-713
18. Magdziarz M., Weron A., and Weron K. Fractional Fokker-Planck dynamics: stochastic representation and computer simulation. Phys Rev E 75 (2007) 016708
19. Maxwell R.M., Welty C., and Harvey R.W. Revisiting the Cape Cod bacteria injection experiment using a stochastic modeling approach. Environ Sci Technol 41 (2007) 5548-5558
20. Meerschaert M.M., Benson D.A., and Bäumer B. Multidimensional advection and fractional dispersion. Phys Rev E 59 5 (1999) 5026-5028 10.1103/PhysRevE.59.5026
21. Meerschaert M.M., and Scheffler H.-P. Limit theorems for continuous time random walks with infinite mean waiting times. J Appl Probab 41 (2004) 623-638
22. Meerschaert M.M., and Scheffler H.-P. Triangular array limits for continuous time random walks. Stoch Proc Appl 118 9 (2008) 1606-1633
23. Michalak A.M., and Kitanidis P.K. Macroscopic behavior and random-walk particle tracking of kinetically sorbing solutes. Water Resour Res 36 8 (2000) 2133-2146
24. Nolan J.P. Numerical calculation of stable densities and distribution functions, heavy tails and highly volatile phenomena. Commun Statist Stoch Models 13 4 (1997) 759-774
25. Painter S., Cvetkovic V., Mancillas J., and Pensado O. Time domain particle tracking methods for simulating transport with retention and first-order transformation. Water Resour Res 44 (2008) W01406 10.1029/2007WR005944
26. Salamon P., Fernàndez-Garcia D., and Gómez-Hernández J.J. A review and numerical assessment of the random walk particle tracking method. J Contam Hydrol 87 (2006) 277-305
27. Salamon P., Fernàndez-Garcia D., and Gòmez-Hernàndez J.J. Modeling mass transfer processes using random walk particle tracking. Water Resour Res 42 (2006) W11417 10.1029/2006WR004927
28. Tompson A.F.B., and Gelhar L.W. Numerical simulation of solute transport in three-dimensional, randomly heterogeneous porous media. Water Resour Res 26 10 (1990) 2541-2562
29. Toride N, Leij FJ, van Genuchten MTh. The CXTFIT code for estimating transport parameters from laboratory or field tracer experiments, v. 2.0. Research Report No. 137. US Salinity Laboratory, U.S.D.A., A.R.S. Riverside, CA, 1995.
30. Schumer R., Benson D.A., Meerschaert M.M., and Baeumer B. Multiscaling fractional advection-dispersion equations and their solutions. Water Resour Res 39 1 (2003) 1022 10.1029/2001WR001229
31. Schumer R., Benson D.A., Meerschaert M.M., and Baeumer B. Fractal mobile/immobile solute transport. Water Resour Res 39 10 (2003) 1296 10.1029/2003WR002141
32. Valocchi AJ, Quinidoz HAM. Application of the random walk method to simulate the transport of kinetically adsorbing solutes. In: Abriola LM, editor. Groundwater contamination. IAHS Publ. No. 185, 1989. p. 35-42.
33. van Genuchten M.T., and Wierenga P.J. Mass transfer studies in sorbing porous media, 1. Analytical solutions. Soil Sci Soc Am J 40 (1976) 473-480
34. Weissmann G.S., Zhang Y., LaBolle E.M., and Fogg G.E. Dispersion of groundwater age in alluvial aquifer system. Water Resour Res 38 10 (2002) 10.1029/2001WR000907 Art. No. 1198
35. Zhang Y., Benson D.A., Meerschaert M.M., and Scheffler H.P. On using random walks to solve the space-fractional advection-dispersion equations. J Stat Phys 123 1 (2006) 89-110
36. Zhang Y., Benson D.A., Meerschaert M.M., LaBolle E.M., and Scheffler H.P. Random walk approximation of fractional-order multiscaling anomalous diffusion. Phys Rev E 74 (2006) 026706
37. Zhang Y., Benson D.A., and Baeumer B. Predicting the tails of breakthrough curves in regional-scale alluvial systems. Ground Water 45 4 (2007) 473-484 10.1111/j.1745-6584.2007.00320.x
38. Zhang Y., and Benson D.A. Lagrangian simulation of multidimensional anomalous transport at the MADE site. Geophys Res Lett 35 (2008) L07403 10.1029/2008GL033222