Nicolae Suciu

Generalized random walk algorithm for the numerical modeling of complex diffusion processes

A generalized form of the random walk algorithm to simulate diffusion processes is introduced. Unlike the usual approach, at a given time all the particles from a grid node are simultaneously scattered using the Bernoulli repartition. This procedure saves memory and computing time and no restrictions are imposed for the maximum number of particles to be used in simulations. We prove that for simple diffusion the method generalizes the finite difference scheme and gives the same precision for large enough number of particles. As an example, simulations of diffusion in random velocity field are performed and the main features of the stochastic mathematical model are numerically tested. 2003 Elsevier Science B.V. All rights reserved.

On the self-averaging of dispersion for transport in quasi-periodic random media

In this study we present a numerical analysis for the self-averaging of the longitudinal dispersion coefficient for transport in heterogeneous media. This is done by investigating the mean-square sample-to-sample fluctuations of the dispersion for finite times and finite numbers of modes for a random field using analytical arguments as well as numerical simulations. We consider transport of point-like injections in a quasi-periodic random field with a Gaussian correlation function. In particular, we focus on the asymptotic and pre-asymptotic behaviour of the fluctuations with the aid of a probability density function for the dispersion, and we verify the logarithmic growth of the sample-to-sample fluctuations as earlier reported in Eberhard (2004 J. Phys. A: Math. Gen. 37 2549–71). We also comment on the choice of the relevant parameters to generate quasi-periodic realizations with respect to the self-averaging of transport in statistically homogeneous Gaussian velocity fields.

A coupled finite element-global random walk approach to advection-dominated transport in porous media with random hydraulic conductivity

Solute transport through heterogeneous porous media considered in environmental and industrial problems is often characterized by high Peclet numbers. In this paper we develop a new numerical approach to advection-dominated transport consisting of coupling an accurate mass-conservative mixed finite element method (MFEM), used to solve Darcy flows, with a particle method, stable and free of numerical diffusion, for non-reactive transport simulations. The latter is the efficient global random walk (GRW) algorithm, which performs the simultaneous tracking of arbitrarily large collections of particles on regular lattices at computational costs comparable to those of single-trajectory simulations using traditional particle tracking (PT). MFEM saturated flow solutions are computed for spatially heterogeneous hydraulic conductivities parameterized as samples of random fields. The coupling is achieved by projecting the velocity field from the MFEM basis onto the regular GRW lattice. Preliminary results show that MFEM-GRW is tens of times faster than the full MFEM flow and transport simulation.

A Fokker-Planck approach for probability distributions of species concentrations transported in heterogeneous media

We identify sufficient conditions under which evolution equations for probability density functions (PDF) of random concentrations are equivalent to Fokker-Planck equations. The novelty of our approach is that it allows consistent PDF approximations by densities of computational particles governed by Ito processes in concentration-position spaces. Accurate numerical solutions are obtained with a global random walk (GRW) algorithm, stable, free of numerical diffusion, and insensitive to the increase of the total number of computational particles. The system of Ito equations is specified by drift and diffusion coefficients describing the PDF transport in the physical space, provided by up-scaling procedures, as well as by drift and mixing coefficients describing the PDF transport in concentration spaces. Mixing models can be obtained similarly to classical PDF approaches or, alternatively, from measured or simulated concentration time series. We compare their performance for a GRW-PDF numerical solution to a problem of contaminant transport in heterogeneous groundwater systems.

Consistency issues in PDF methods

Abstract Concentrations of chemical species transported in random environments need to be statistically characterized by probability density functions (PDF). Solutions to evolution equations for the one-point one-time PDF are usually based on systems of computational particles described by Itô equations. We establish consistency conditions relating the concentration statistics to that of the Itô process and the solution of its associated Fokker-Planck equation to that of the PDF equation. In this frame, we use a recently proposed numerical method which approximates PDFs by particle densities obtained with a global random walk (GRW) algorithm. The GRW-PDF approach is illustrated for a problem of contaminant transport in groundwater.

Ergodic Simulations for Diffusion in Random Velocity Fields

Ergodic simulations aim at estimating ensemble average characteristics of diffusion in random fields from space averages. The traditional approach, based on large supports of the initial concentration in general fails to obtain ergodic simulations. However, such simulations, using single realizations of the velocity, are shown to be feasible if space averages with respect to the location of the initial concentration support are used to estimate ensemble averages.

Solute transport in aquifers with evolving scale heterogeneity

Abstract Transport processes in groundwater systems with spatially heterogeneous properties often exhibit anomalous behavior. Using first-order approximations in velocity fluctuations we show that anomalous superdiffusive behavior may result if velocity fields are modeled as superpositions of random space functions with correlation structures consisting of linear combinations of short-range correlations. In particular, this corresponds to the superposition of independent random velocity fields with increasing integral scales proposed as model for evolving scale heterogeneity of natural porous media [Gelhar, L. W. Water Resour. Res. 22 (1986), 135S-145S]. Monte Carlo simulations of transport in such multi-scale fields support the theoretical results and demonstrate the approach to superdiffusive behavior as the number of superposed scales increases.

Comment on “Spatial moments analysis of kinetically sorbing solutes in aquifer with bimodal permeability distribution” by M. Massabó, A. Bellin, and A. J. Valocchi

[1] Massabó et al. [2008] derived useful first-order estimations of ergodic dispersivities for transport of kinetically sorbing solutes in heterogeneous porous media with bimodal hydraulic conductivity distributions. Such dispersivity estimates can serve as comparison terms and, in case of infinite domains, they also can be used as input parameters of transport operator for multicomponent transport-reaction problems in heterogeneous porous media [Kräutle and Knabner, 2005, 2007]. In this commentary we try to establish conditions for the applicability of the ergodic parameters obtained by Massabó et al. [2008] to real cases corresponding to single-realizations of the stochastic models. [2] Massabó et al. [2008, paragraph 20] claim that ‘‘a well established result of stochastic theories is that for plumes with transverse dimension much larger than the corresponding integral scale, transport develops under ergodic conditions’’ and the single realization dispersion slm can be approximated by the sum between the second moment of the initial plume Slm(0) and the one-particle dispersion Xlm, where l and m range between 1 and the dimensionality of the problem:

Ergodic Estimations of Upscaled Coefficients for Diffusion in Random Velocity Fields

Upscaled coefficients for diffusion in ergodic velocity fields are derived by summing up correlations of increments of the position process, or equivalently of the Lagrangian velocity. Ergodic estimations of the correlations are obtained from time averages over finite paths sampled on a single trajectory of the process and a space average with respect to the initial positions of the paths. The first term in this path decomposition of the diffusion coefficients corresponds to Markovian diffusive behavior and is the only contribution for processes with independent increments. The next terms describe memory effects on diffusion coefficients until they level off to the value of the upscaled coefficients. Since the convergence with respect to the path length is rather fast and no repeated Monte Carlo simulations are required, this method speeds up the computation of the upscaled coefficients over methods based on long-time limit and ensemble averages by four orders of magnitude.

Global Random Walk Simulations of Diffusion

Random walk methods are suitable to build up convergent solutions for reaction-diffusion problems and were successfully applied to simulations of transport processes in a random environment. The disadvantage is that, for realistic cases, these methods become time and memory expensive. To increase the computation speed and to reduce the required memory, we derived a “global random walk” method in which the particles at a given site of the grid are simultaneously scattered following the binomial Bernoulli repartition. It was found that the computation time is reduced three orders of magnitude with respect to individual random walk methods. Moreover, by suitable “microscopic balance” boundary conditions, we obtained good simulations of transport in unbounded domains, using normal size grids. The global random walk improves the statistical quality of simulations for diffusion processes in random fields. The method was tested by comparisons with analytical and finite difference solutions as well as with concentrations measured in “column experiments”, used in laboratory study of soils’ hydrogeological and chemical properties.