# 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

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## 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

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##### Journal

Monte Carlo Methods and Applications

0929-9629

1569-3961