A coupled finite element-global random walk approach to advection-dominated transport in porous media with random hydraulic conductivity - Suciu, Nicolae; Radu, Adrian Florin; Prechtel, Alexander; Brunner, Fabian; Knabner, Peter (Journal of Computational and Applied Mathematics, 2013)
Solute transport through heterogeneous porous media considered in environmental and industrial problems is often characterized by high Péclet 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. © 201
Keywords: Advection-dominated transport; Porous media; Random walk methods

A Fokker-Planck approach for probability distributions of species concentrations transported in heterogeneous media - Suciu, Nicolae; Radu, Adrian Florin; Attinger, Sabine; Schüler, Lennart; Knabner, Peter (Journal of Computational and Applied Mathematics, 2015)
Abstract 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 Itô 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 Itô 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 system
Keywords: Mixing; PDF methods; Porous media; Random walk

A time dependent mixing model to close PDF equations for transport in heterogeneous aquifers - Schüler, Lennart; Suciu, Nicolae; Knabner, Peter; Attinger, Sabine (Advances in Water Resources, 2016)
Probability density function (PDF) methods are a promising alternative to predicting the transport of solutes in groundwater under uncertainty. They make it possible to derive the evolution equations of the mean concentration and the concentration variance, used in moment methods. The mixing model, describing the transport of the PDF in concentration space, is essential for both methods. Finding a satisfactory mixing model is still an open question and due to the rather elaborate PDF methods, a difficult undertaking. Both the PDF equation and the concentration variance equation depend on the same mixing model. This connection is used to find and test an improved mixing model for the much easier to handle concentration variance. Subsequently, this mixing model is transferred to the PDF equation and tested. The newly proposed mixing model yields significantly improved results for both variance modelling and PDF modellin
Keywords: Global random walk; Heterogeneity; Mixing; PDF method; Solute transport; Variance

Biased Global Random Walk, a Cellular Automaton for Diffusion - Suciu, Nicolae; Vamos, Calin; Knabner, Peter; Rüde, Ulrich (Simulationstechnique. 18th Symposium in Erlangen. September 2005, 2005)
UnivIS-Import:2015-04-16:Pub.2005.tech.IMMD.lsinfs.biased

Comment on "Spatial moments analysis of kinetically sorbing solutes in aquifer with bimodal permeability distribution" by M. Massabó, A. Bellin, and A. J. Valocchi - Suciu, Nicolae; Knabner, Peter (Water Resources Research, 2009)

Consistency issues in pdf methods - Suciu, Nicolae; Schüler, Lennart; Radu, Adrian Florin; Attinger, Sabine; Vamo̧s, Cǎlin; Knabner, Peter (Analele Stiintifice ale Universitatii Ovidius Constanta, Seria Matematica, 2015)
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 describee by Ito equations. We establish consistency conditions relating the concentration statistics to that of the Ito 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 groundwate
Keywords: Mixing; PDF methods; Porous media; Random walk

Diffusion in Random Fields - Applications to Transport in Groundwater - Suciu, Nicolae (Birkhäuser, 2019)
This book presents, in an accessible and self-consistent way, the theory of diffusion in random velocity fields, together with robust numerical simulation approaches. The focus is on transport processes in natural porous media, with applications to contaminant transport in groundwater. Starting from basic information on stochastic processes, more challenging issues are subsequently addressed, such as the correlation structure of the diffusion process in random fields, the relation between memory effects and ergodic properties, derivation and parameterizations of evolution equations for probability densities, and the relation between measurements and spatio-temporal upscaling. Written for readers with a background in applied mathematics, engineering, physics or geophysics, the book offers an essential basis for further research in the stochastic modeling of groundwater systems.<br

Diffusion in random velocity fields with applications to contaminant transport in groundwater - Suciu, Nicolae (Advances in Water Resources, 2014)
The process of diffusion in a random velocity field is the mathematical object underlying currently used stochastic models of transport in groundwater. The essential difference from the normal diffusion is given by the nontrivial correlation of the increments of the process which induces transitory or persistent dependence on initial conditions. Intimately related to these memory effects is the ergodicity issue in subsurface hydrology. These two topics are discussed here from the perspectives of Itô and Fokker–Planck complementary descriptions and of recent Monte Carlo studies. The latter used a global random walk algorithm, stable and free of numerical diffusion. Beyond Monte Carlo simulations, this algorithm and the mathematical frame of the diffusion in random fields allow efficient solutions to evolution equations for the probability density of the random concentration.<br
Keywords: Groundwater; Transport processes; Ergodicity; Random fields; Random walk; PDF methods

Global Random Walk simulations for sensitivity and uncertainty analysis of passive transport models - Suciu, Nicolae; Vamo̧s, Cǎlin; Vereecken, Harry; Knabner, Peter (Mathematics and its Applications : Annals of the Academy of Romanian Scientists, 2011)
The Global Random Walk algorithm (GRW) performs a simultaneous tracking on a fixed grid of huge numbers of particles at costs comparable to those of a single-trajectory simulation by the traditional Particle Tracking (PT) approach. Statistical ensembles of GRW simulations of a typical advection-dispersion process in groundwater systems with randomly distributed spatial parameters are used to obtain reliable estimations of the input parameters for the upscaled transport model and of their correlations, input-output correlations, as well as full probability distributions of the input and output parameter
Keywords: Groundwater contamination; Monte carlo methods; Probabilistic particle methods; Transport processes

Global Random Walk Solutions for Flow and Transport in Porous Media - Suciu, Nicolae (Lecture Notes in Computational Science and Engineering, 2021)
This article presents a new approach to solve the equations of flow in heterogeneous porous media by using random walks on regular lattices. The hydraulic head is represented by computational particles which are spread globally from the lattice sites according to random walk rules, with jump probabilities determined by the hydraulic conductivity. The latter is modeled as a realization of a random function generated as a superposition of periodic random modes. One- and two-dimensional numerical solutions are validated by comparisons with analytical manufactured solutions. Further, an ensemble of divergence-free velocity fields computed with the new approach is used to conduct Monte Carlo simulations of diffusion in random fields. The transport equation is solved by a global random walk algorithm which moves computational particles representing the concentration of the solute on the same lattice as that used to solve the flow equations. The integrated flow and transport solution is validated by a good agreement between the statistical estimations of the first two spatial moments of the solute plume and the predictions of the stochastic theory of transport in groundwater.
CRIS-Team Scopus Importer:2021-06-04

Global random walk solvers for fully coupled flow and transport in saturated/unsaturated porous media - Suciu, Nicolae; Illiano, Davide; Prechtel, Alexander; Radu, Adrian Florin (Advances in Water Resources, 2021)
In this article, we present new random walk methods to solve flow and transport problems in saturated/unsaturated porous media, including coupled flow and transport processes in soils, heterogeneous systems modeled through random hydraulic conductivity and recharge fields, processes at the field and regional scales. The numerical schemes are based on global random walk algorithms (GRW) which approximate the solution by moving large numbers of computational particles on regular lattices according to specific random walk rules. To cope with the nonlinearity and the degeneracy of the Richards equation and of the coupled system, we implemented the GRW algorithms by employing linearization techniques similar to the L-scheme developed in finite element/volume approaches. The resulting GRW Lschemes converge with the number of iterations and provide numerical solutions that are first-order accurate in time and second-order in space. A remarkable property of the flow and transport GRW solutions is that they are practically free of numerical diffusion. The GRW solvers are validated by comparisons with mixed finite element and finite volume solvers in one- and two-dimensional benchmark problems. They include Richards' equation fully coupled with the advection-diffusion-reaction equation and capture the transition from unsaturated to saturated flow regimes.<br
Keywords: Richards equation; Coupled flow and transport; Linearization; Iterative schemes; Global random walk

Global random walk solvers for reactive transport and biodegradation processes in heterogeneous porous media - Suciu, Nicolae; Radu, Florin A. (Advances in Water Resources, 2022)
Flow and multicomponent reactive transport in saturated/unsaturated porous media are modeled by ensembles of computational particles moving on regular lattices according to specific random walk rules. The occupation number of the lattice sites is updated with a global random walk (GRW) procedure which spreads the particles from a lattice site with computational costs comparable to those for a single random walk step in sequential procedures. To cope with the nonlinearity and the degeneracy of the Richards equation the GRW flow solver uses linearization techniques similar to the L-scheme developed in finite element/volume approaches. Numerical schemes for reactive transport, coupled with the flow solver via numerical solutions for saturation and water flux, are implemented in splitting procedures. Diffusion-advection steps are solved by GRW algorithms using either biased or unbiased random walk probabilities. Since the number of particles in GRW simulations can be as large as the number of molecules involved in chemical reactions, one avoids the cumbersome problem of rescaling particle densities to approximate concentrations. Reaction steps are therefore formulated in terms of concentrations, as in deterministic approaches. The numerical convergence of the new schemes is demonstrated by comparisons with manufactured analytical solutions. Coupled flow and reactive transport problems of contaminant biodegradation described by the Monod model are further solved and the influence of flow nonlinearity/degeneracy and of the spatial heterogeneity of the medium is investigated numerically.
CRIS-Team WoS Importer:2022-10-14

Ito equation model for dispersion of solutes in heterogeneous media - Suciu, Nicolae; Vamo̧s, Cǎlin; Vereecken, Harry; Sabelfeld, Karl; Knabner, Peter (Revue d'Analyse Numérique et de Théorie de l'Approximation, 2008)
Transport processes in heterogeneous media such as ionized plasmas, natural porous media, and turbulent atmosphere are often modeled as diusion processes in random velocity elds. Using the Ito ̂ formalism, we decompose the second spatial moments of the concentration and the equivalent eective dispersion coecients in terms corresponding to various physical factors which in uence the transport. We explicitly dene " dispersion coecients, independent of the initial conditions and completely determined by local disper-sion coecients and velocity correlations. Ergodic coecients govern an upscaled process which describes the transport at large tine-space scales. The non-ergodic behavior at nite times shown by numerical experiments for large initial plumes is explained by terms " accounting for correlations between initial po-sitions and velocity uctuations on the trajectories of the solute molecule
Keywords: second spatial moment; solute molecule; upscaled process; memory term; ergodic dispersion coecients; nite time; heterogeneous medium; initial positions; ito formalism; ergodic coecients; non-ergodic behavior; large tine-space scale; velocity correlation; equivalent eective dispersion; coecients; initial condition; natural porous medium; large initial plume; various physical factor; local disper-sion coecients; numerical experiment; turbulent atmosphere; random velocity eld; velocity uctuations; ionized plasma

Memory effects induced by dependence on initial conditions and ergodicity of transport in heterogeneous media - Suciu, Nicolae; Vamo̧s, Cǎlin; Vereecken, Harry; Sabelfeld, Karl; Knabner, Peter (Water Resources Research, 2008)
For transport in statistically homogeneous random velocity fields with properties that are routinely assumed in stochastic groundwater models, the one-particle dispersion (i.e., second central moment of the ensemble average concentration for a point source) is a "memory-free" quantity independent of initial conditions. Nonergodic behavior of large initial plumes, as manifest in deviations of actual solute dispersion from one-particle dispersion, is associated with a "memory term" consisting of correlations between initial positions and displacements of solute molecules. Reliable numerical experiments show that increasing the source dimensions has two opposite effects: it reduces the uncertainty related to the randomness of center of mass, but, at the same time, it yields large memory terms. The memory effects increase with the source dimension and depend on its shape and orientation. Large narrow sources oriented transverse to the mean flow direction yield ergodic behavior with respect to the one-particle dispersion of the longitudinal dispersion and nonergodic behavior of the transverse dispersion, whereas for large longitudinal sources, the longitudinal dispersion behaves nonergodically, and the transverse dispersion behaves ergodically. Such memory effects are significantly large over hundreds of heterogeneity scales and should therefore be considered in practical applications, for instance, calibration of model parameters, forecasting, and identification of the contaminant source. Copyright 2008 by the American Geophysical Unio

Numerical benchmark study for flow in highly heterogeneous aquifers - Alecsa, Cristian D.; Boros, Imre; Frank, Florian; Knabner, Peter; Nechita, Mihai; Prechtel, Alexander; Rupp, Andreas; Suciu, Nicolae (Advances in Water Resources, 2020)
This article presents numerical investigations on accuracy and convergence properties of several numerical approaches for simulating steady state flows in heterogeneous aquifers. Finite difference, finite element, discontinuous Galerkin, spectral, and random walk methods are tested on two-dimensional benchmark flow problems. Realizations of log-normal hydraulic conductivity fields are generated by Kraichnan algorithms in closed form as finite sums of random periodic modes, which allow direct code verification by comparisons with manufactured reference solutions. The quality of the methods is assessed for increasing number of random modes and for increasing variance of the log-hydraulic conductivity fields with Gaussian and exponential correlation. Experimental orders of convergence are calculated from successive refinements of the grid. The numerical methods are further validated by comparisons between statistical inferences obtained from Monte Carlo ensembles of numerical solutions and theoretical first-order perturbation results. It is found that while for Gaussian correlation of the log-conductivity field all the methods perform well, in the exponential case their accuracy deteriorates and, for large variance and number of modes, the benchmark problems are practically not solvable with reasonably large computing resources, for all the methods considered in this study.<br

Persistent memory of diffusing particles - Suciu, Nicolae; Vamo̧s, Cǎlin; Radu, Adrian Florin; Vereecken, Harry; Knabner, Peter (Physical Review E, 2009)
The variance of the advection-diffusion processes with variable coefficients is exactly decomposed as a sum of dispersion terms and memory terms consisting of correlations between velocity and initial positions. For random initial conditions, the memory terms quantify the departure of the preasymptotic variance from the time-linear diffusive behavior. For deterministic initial conditions, the memory terms account for the memory of the initial positions of the diffusing particles. Numerical simulations based on a global random walk algorithm show that the influence of the initial distribution of the cloud of particles is felt over hundreds of dimensionless times. In case of diffusion in random velocity fields with finite correlation range the particles forget the initial positions in the long-time limit and the variance is self-averaging, with clear tendency toward normal diffusion. © 2009 The American Physical Societ

Solute transport in aquifers with evolving scale heterogeneity - Suciu, Nicolae; Attinger, Sabine; Radu, Adrian Florin; Vamo̧s, Cǎlin; Vanderborght, Jan; Vereecken, Harry; Knabner, Peter (The Preprint-Series of the Institute of Applied Mathematics, 2011)
Transport processes in groundwater systems with spatially heterogeneous properties often exhibit anomalous behavior. Using first-order approximations in velocity fluctuations we show that anomalous superdif-fusive 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 increase
Keywords: Porous media; Random fields; Random walk; Transport

Solute transport in aquifers with evolving scale heterogeneity - Suciu, Nicolae; Attinger, Sabine; Radu, Adrian Florin; Vamo̧s, Cǎlin; Vanderborght, Jan; Vereecken, Harry; Knabner, Peter (Analele Stiintifice ale Universitatii Ovidius Constanta, Seria Matematica, 2015)
Transport processes in groundwater systems with spatially heterogeneous properties often exhibit anomalous behavior. Using first-order approximations in velocity fluctuations we show that anomalous superdif-fusive 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 increase
Keywords: Porous media; Random fields; random walk; transport

Towards a filtered density function approach for reactive transport in groundwater - Suciu, Nicolae; Schüler, Lennart; Attinger, Sabine; Knabner, Peter (Advances in Water Resources, 2016)
Evolution equations for probability density functions (PDFs) and filtered density functions (FDFs) of random species concentrations weighted by conserved scalars are formulated as Fokker-Planck equations describing stochastically equivalent processes in concentration-position spaces. This approach provides consistent numerical PDF/FDF solutions, given by the density in the concentration-position space of an ensemble of computational particles governed by the associated Itô equations. The solutions are obtained by a global random walk (GRW) algorithm, which is stable, free of numerical diffusion, and practically insensitive to the increase of the number of particles. The general FDF approach and the GRW numerical solution are illustrated for a reduced complexity problem consisting of the transport of a single scalar in groundwater. Randomness is induced by the stochastic parameterization of the hydraulic conductivity, characterized by short range correlations and small variance. The objective is to infer the statistics of the random concentration sampled at the plume center of mass, integrated over the transverse dimension of a two-dimensional spatial domain. The PDF/FDF problem can therefore be formulated in a two-dimensional domain as well, a spatial dimension and one in the concentration space. The upscaled drift and diffusion coefficients describing the PDF transport in the physical space are estimated on single-trajectories of diffusion in velocity fields with short-range correlations, owing to their self-averaging property. The mixing coefficients describing the PDF transport in concentration spaces are parameterized by the trend and the noise inferred from the statistical analysis of an ensemble of simulated concentration time series, as well as by classical mixing models. A Gaussian spatial filter applied to a Kraichnan velocity field generator is used to construct coarse-grained simulations (CGS) for FDF problems. The purposes of the CGS simulations are two-fold: first to understand the significance of the FDF approach from a practical point of view and its relation to the PDF approach; second to investigate the limits of the mixing models considered here and the desirable features of the mixing models for groundwater system
Keywords: Global random walk; Groundwater; Mixing; PDF/FDF methods