Numerical Modeling of Large Scale Transport of Contaminant Solutes Using the Global Random Walk Algorithm

Abstract

The methods which track particles to simulate concentrations are successfully used in problems for large scale transport in groundwater, mainly when the aquifer properties are spatially heterogeneous and the process is advection dominated. These methods, sometimes called “analogical Monte Carlo methods”, are not concerned with the numerical diffusion occurring in finite difference/element schemes. The limitations of classical random walk methods are due to large computation time and memory necessary to achieve statistically reliable results and accurate concentration fields. To overcome these computational limitations a “global random walk” (GRW) algorithm was developed. Unlike in the usual approach where the trajectory of each particle is simulated and stored, in GRW all the particles from a given grid node are scattered, at a given time, using a single numerical procedure, based on Bernoulli distribution, partial-deterministic, or deterministic rules. Because no restrictions are imposed for the maximum number of particles to be used in a simulation, the Monte Carlo repetitions are no longer necessary to achieve the convergence. It was proved that for simple diffusion problems GRW converges to the finite difference scheme and that for large scale transport problems in groundwater, GRW produces stable and statistically reliable simulations. A 2-dimensional transport problem was modeled by simulating local diffusion processes in realizations of a random velocity field. The behavior over 5000 days of the effective diffusion coefficients, concentration field and concentration fluctuations were investigated using 2560 realizations of the velocity field and 1010 particles in every realization. The results confirm the order of magnitude of the effective diffusion coefficients predicted by stochastic theory but the time needed to reach the asymptotic regime was found to be thousands times larger. It is also underlined that the concentration fluctuations and the dilution of contaminant solute depend essentially on local diffusion and boundary conditions.

 

Authors

N. Suciu
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

C. Vamoş
Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy

H. Hardelauf
Research Center Julich (Germany), ICG-IV: Institut of Agrosphere

J. Vanderborght
Research Center Julich (Germany), ICG-IV: Institut of Agrosphere

H. Vereecken
Research Center Julich (Germany), ICG-IV: Institut of Agrosphere

Keywords

?

Cite this paper as

N. Suciu, C. Vamoş, J. Vanderborght, H. Hardelauf, H. Vereecken (2004), Numerical modeling of large scale transport of contminant solutes using the global random walk algorithm, Monte Carlo Methods and Applications, 10(2), 153-177, doi: 10.1163/156939604777303235

References

see the expansion block below.

PDF

https://s3.amazonaws.com/academia.edu.documents/45765999/MCMA2004.pdf?AWSAccessKeyId=AKIAIWOWYYGZ2Y53UL3A&Expires=1534854817&Signature=0oS6UoxpA34HJOqVVoxUQjQY4c4%3D&response-content-disposition=inline%3B%20filename%3DNumerical_Modeling_of_Large_Scale_Transp.pdf

soon

About this paper

Journal

Monte Carlo Methods and Applications

Publisher Name
Print ISSN

0929-9629

Online ISSN

1569-3961

Google Scholar Profile

google scholar profile

References

Paper in html format

References

[1] Ames, W. F., Numerical Methods for Partial Differential Equations, 2nd ed., Academic Press, New York, 1977.

[2] Bellin, A., P. Salandin and A. Rinaldo, Simulation of dispersion in heterogeneous porous formations: Statistics, first-order theories, convergence of computations, Water Resour. Res. 28(9), 2211-2227, 1992.
CrossRef (DOI)

[3] Chorin, A. J., Vortex sheet approximation of boundary layers, J. Comput. Phys. 27, 428-442 (1978).
CrossRef (DOI)

[4] Dagan, G., Flow and Transport in Porous Formations, Springer, New York, 1989.
CrossRef (DOI)

[5] Dagan, G. and A. Fiori, The influence of pore-scale dispersion on concentration statistical moments in transport through heterogeneous aquifers, Water Resour. Res., 33, 1595-1650, 1997.
CrossRef (DOI)

[6] Dentz, M., H. Kinzelbach, S. Attinger, and W. Kinzelbach, Temporal behavior of a solute cloud in a heterogeneous porous medium 3. Numerical simulations, Water Resour. Res., 38, 23:1-12, 2002.
CrossRef (DOI)

[7] Dentz, M., H. Kinzelbach, S. Attinger, and W. Kinzelbach, Numerical studies of the transport behavior of a passive solute in a two-dimensional incompressible random flow field, Phys. Rev. E 67, 046306:1-10, 2003.
CrossRef (DOI)

[8] Fiori, A., Finite Peclet extension of Dagan’s solutions to transport in anisotropic formations, Water Resour. Res., 32, 193-198, 1996.
CrossRef (DOI)

[9] Kapoor, C. and L. W. Gelhar, Transport in three-dimensionallity heterogeneous aquifers 1. Dynamics of concentration fluctuations, Water Resour. Res., 30, 1775-1789, 1994.
CrossRef (DOI)

[10] Kapoor, C. and P. K. Kitanidis, Concentration fluctuation and dilution in aquifers, Water Resour. Res., 34, 1181-1193, 1998.
CrossRef (DOI)

[11] Lundgren, T. S., Turbulent pair dispersion and scalar diffusion, J. Fluid Mech., 111, 27-57, 1981.
CrossRef (DOI)

[12] Pannone, M. and P. K. Kitanidis, Large-time behavior of concentration variance and dilution in heterogeneous media, Water Resour. Res., 35, 623-634, 1999.
CrossRef (DOI)

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

[14] Sabelfeld, K. K. and G. A. Mikhailov, On the numerical simulation of the particle’s diffusion in random velocity fields, Izv. AN SSSR, Physics of Atmosphere and Ocean, 16 (3), 229-235 (in Russian), 1980.

[15] Sabelfeld, K. K., Monte Carlo methods in boundary value problems, Springer, New York – Heidelberg – Berlin, 1991.

[16] Schwarze, H., U. Jaekel and H. Vereecken, Estimation of macrodispersivity by different approximation methods for flow and transpot in randomly heterogeneous media, Transport in Porous Media 43, 265-287, 2001.
CrossRef (DOI)

[17] Suciu, N., C. Vamos, H. Vereecken, and J.Vanderborght, pp. 111-140 in G. Marinovschi and I. Stelian (Eds.) Proceedings of the first Workshop on Mathematical Modeling of Environmental Problems, Rom. Acad. Publishing House, Bucharest 2002.
CrossRef (DOI)

[18] Suciu N., C. Vamos, H. Vereecken, J. Vanderborght, and H. Hardelauf, pp. 215-221 in T. Maghiar, A. Georgescu, M. Balaj, I. Dzitac, I. Mang (Eds.), Proc. of the 11th Conf. of Appl. and Ind. Math., Vol. 1, Oradea Univ. Publishing House, Oradea, 2003.

[19] Suciu, N. and C. Vamos, Effective diffusion in heterogeneous media, Interner Bericht ICG-IV 00303, Forschungszentrum Julich, 2003. Sun, Ne-Z., Mathematical Modeling in Groundwater Pollution, Springer, New York, 1996.

[20] Vamos, C., N. Suciu, H. Vereecken, J. Vanderborht and O. Nitzsche, Path decomposition of discrete effective diffusion coefficient, Internal Report ICG-IV.00501, Forschungszentrum Julich, 2001.

[21] Vamo¸s, C., N. Suciu, and H. Vereecken, Generalized random walk algorithm for the numerical modeling of complex diffusion processes, J. Comp. Phys., 186, 527-544, 2003.
CrossRef (DOI)

[22] van Kampen, N. G., Stochastic Processes in Physics and Chemistry, North-Holland, Amsterdam, 1981.

Related Posts

Menu