Global random walk modeling of transport in complex systems

Abstract

The Global random walk algorithm performs simultaneously the tracking of large collections of particles and permits massive simulations at reasonable costs. Applications were developed for transport in systems with anisotropic, non-homogeneous, and randomly distributed parameters. As a first illustration we present simulations for diffusion in human skin. Further, a case study for contaminant transport in groundwater shows that the realizations of the transport process converge in mean square limit to a Gaussian diffusion. This investigation also indicates that the use of the Kraichnan routine, based on periodic random fields, yields reliable simulations of transport in Gaussian velocity fields.

Authors

N. Suciu
Tiberiu Popoviciu Institute of Numerical Analysis, Cluj Napoca branch of the Romanian Academy

C. Vamoş
Tiberiu Popoviciu, Institutue of Numerical and analysis, Romanian Academiy

I. Turcu
National R&D Institute for Isotopic and Molecular Technologies, Cluj-Napoca, Romania

C.V.L. Pop
National R&D Institute for Isotopic and Molecular Technologies, Cluj-Napoca, Romania

L.I. Ciortea
National R&D Institute for Isotopic and Molecular Technologies, Cluj-Napoca, Romania

Keywords

Global random walk; lattice gas; human skin; groundwater contamination

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Cite this paper as:

N. Suciu, C. Vamoş, I. Turcu, C.V.L. Pop, L.I. Ciortea (2007), Global random walk modeling of transport in complex systems, Computing and Visualization in Science, doi: 10.1007/s00791-007-0077-6

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  • [1] Eberhard, J., Approximations for transport parameters and selfaveraging properties for point-like injections in heterogeneous media. J. Phys. A Math. Gen. 37, 2549–2571 (2004)[2] Eberhard, J., Suciu N., Vamos, On the self-averaging of dispersion for transport in quasi-periodic random media. J. Phys. A: Math. Gen. 40, 597–610, doi:10.1088/1751-8113/40/4/002 (2007)
    [3] Jaekel, U., Vereecken, H.: Renormalization group analysis of macrodispersion in a directed random flow. Water Resour. Res. 33, 2287–2299 (1997)

    [4] Karapiperis, T., Blankleider, B., Cellular automaton model of reaction-transport processes. Physica D 78, 30–64 (1991)

    [5] Kesten, H., Papanicolaou, G.C., A limit theorem for turbulent diffusion. Commun. Math. Phys. 65, 97–128 (1979)

    [6] Kinzelbach, W., Uffink, G. , The random walk method and extensions in groundwater modelling. In: Bear, J., Corapcioglu, M.Y. (eds.) Transport Processes in Porous Media, pp. 761–787. Kluwer, Norwell (1991)

[7] Johnson, M.E., Blankschtein, D., Langer, R., Evaluation of solute permeation through the stratum corneum: lateral bilayer diffusion as the primary transport mechanism. J. Pharm. Sci. 86(10), 1162– 1172 (1997)

[8] Roth, K., Hammel, K., Transport of conservative chemical through an unsaturated two-dimensional Miller-similar medium with steady state flow. Water Resour. Res. 32(6), 1653–1663 (1996)

[9] Schwarze, H., Jaekel, U., Vereecken, H., Estimation of macrodispersivity by different approximation methods for flow and transport in randomly heterogeneous media. Transp. Porous Media 43, 265– 287 (2001)

[10] Suciu, N., Vamos, C., Vanderborght, J., Hardelauf, H., Vereecken, H., Numerical modeling of large scale transport of contminant solutes using the global random walk algorithm. Monte Carlo Methods Appl. 10(2), 153–177 (2004)

[11] Suciu, N., Vamos, C., Knabner, P., Rüde, U., Biased global random walk, a cellular automaton for diffusion. In: Hüsemann, F., Kowarschik, M., Rüde, U. (eds.) Simulationstechnique, 18th Symposium in Erlangen, September 2005., pp. 562–567. SCS Publishing House e. V., Erlangen (2005)

[12] Suciu, N., Vamos, C., Evaluation of overshooting errors in particle methods for diffusion by biased global random walk. Rev. Anal. Num. Th. Approx. 35, 119–126 (2006)

[13] Suciu, N., Vamos, C., Vanderborght, J., Hardelauf, H., Vereecken, H., Numerical investigations on ergodicity of solute transport in heterogeneous aquifers, Water Resour. Res. 42, W04409. doi:10.1029/2005WR004546 (2006)

[14] Suciu, N., Vamos, C., Eberhard, J., Evaluation of the firstorder approximations for transport in heterogeneous media. Water Resour. Res. 42, W11504. doi:10.1029/2005WR004714 (2006)

[15] Vamos, C., Suciu, N., Vereecken, H., Generalized random walk algorithm for the numerical modeling of complex diffusion processes. J. Comput. Phys. 186(2), 527–544 (2003

[16] Vamos, C., Suciu, N., Turcu, I., Pop, C.V.L., Ciortea, L. I.,  Programme BIOTECH, Project No. 01-8-CPD-042/19.10.2001. Research report (2004, in Romanian)

[17] Vereecken, H., Döring, U., Hardelauf, H., Jaekel, U., Hashagen, U., Neuendorf, O., Schwarze, H., Seidemann, R.,  Analysis of solute transport in a heterogeneous aquifer: the Krauthausen field experiment. J. Contam. Hydol. 45, 329–358 (2000)

[18] Wilke, J., Pohl, T., Kowarschik, M., Rüde, U., Cache performance optimizations for parallel lattice Boltzmann codes. In: Proceedings of the EuroPar-03 Conference. Lecture Notes in Computer Science, vol. 2790, pp. 441–450. Springer, Heidelberg (2003)

[1] Eberhard, J., Approximations for transport parameters and selfaveraging properties for point-like injections in heterogeneous media. J. Phys. A Math. Gen. 37, 2549–2571 (2004)

[2] Eberhard, J., Suciu N., Vamos,
On the self-averaging of dispersion for transport in quasi-periodic random media. J. Phys. A: Math. Gen. 40, 597–610, doi:10.1088/1751-8113/40/4/002 (2007)

[3] Jaekel, U., Vereecken, H.,
Renormalization group analysis of macrodispersion in a directed random flow. Water Resour. Res. 33, 2287–2299 (1997)

[4] Karapiperis, T., Blankleider, B.,
Cellular automaton model of reaction-transport processes. Physica D 78, 30–64 (1991)

[5] Kesten, H., Papanicolaou, G.C.,
A limit theorem for turbulent diffusion. Commun. Math. Phys. 65, 97–128 (1979)

[6] Kinzelbach, W., Uffink, G.,
The random walk method and extensions in groundwater modelling. In: Bear, J., Corapcioglu, M.Y. (eds.) Transport Processes in Porous Media, pp. 761–787. Kluwer, Norwell (1991)

[7] Johnson, M.E., Blankschtein, D., Langer, R.,
Evaluation of solute permeation through the stratum corneum: lateral bilayer diffusion as the primary transport mechanism. J. Pharm. Sci. 86(10), 1162– 1172 (1997)

[8] Roth, K., Hammel, K.,
Transport of conservative chemical through an unsaturated two-dimensional Miller-similar medium with steady state flow. Water Resour. Res. 32(6), 1653–1663 (1996)

[9] Schwarze, H., Jaekel, U., Vereecken, H.,
Estimation of macrodispersivity by different approximation methods for flow and transport in randomly heterogeneous media. Transp. Porous Media 43, 265– 287 (2001)

[10] Suciu, N., Vamos, C., Vanderborght, J., Hardelauf, H., Vereecken, H.,
Numerical modeling of large scale transport of contminant solutes using the global random walk algorithm. Monte Carlo Methods Appl. 10(2), 153–177 (2004)

[11] Suciu, N., Vamos, C., Knabner, P., Rüde, U.,
Biased global random walk, a cellular automaton for diffusion. In: Hüsemann, F., Kowarschik, M., Rüde, U. (eds.) Simulationstechnique, 18th Symposium in Erlangen, September 2005., pp. 562–567. SCS Publishing House e. V., Erlangen (2005)

[12] Suciu, N., Vamos, C.,
Evaluation of overshooting errors in particle methods for diffusion by biased global random walk. Rev. Anal. Num. Th. Approx. 35, 119–126 (2006)

[13] Suciu, N., Vamos, C., Vanderborght, J., Hardelauf, H., Vereecken, H.,
Numerical investigations on ergodicity of solute transport in heterogeneous aquifers, Water Resour. Res. 42, W04409. doi:10.1029/2005WR004546 (2006)

[14] Suciu, N., Vamos, C., Eberhard, J.,
Evaluation of the firstorder approximations for transport in heterogeneous media. Water Resour. Res. 42, W11504. doi:10.1029/2005WR004714 (2006)

[15] Vamos, C., Suciu, N., Vereecken, H.,
Generalized random walk algorithm for the numerical modeling of complex diffusion processes. J. Comput. Phys. 186(2), 527–544 (2003)

[16] Vamos, C., Suciu, N., Turcu, I., Pop, C.V.L., Ciortea, L. I.,
Programme BIOTECH, Project No. 01-8-CPD-042/19.10.2001. Research report (2004, in Romanian)

[17] Vereecken, H., Döring, U., Hardelauf, H., Jaekel, U., Hashagen, U., Neuendorf, O., Schwarze, H., Seidemann, R.,
Analysis of solute transport in a heterogeneous aquifer: the Krauthausen field experiment. J. Contam. Hydol. 45, 329–358 (2000)

[18] Wilke, J., Pohl, T., Kowarschik, M., Rüde, U.,
Cache performance optimizations for parallel lattice Boltzmann codes. In: Proceedings of the EuroPar-03 Conference. Lecture Notes in Computer Science, vol. 2790, pp. 441–450. Springer, Heidelberg (2003)

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