Water Resources Research, 2006

1] The complexity of mass transfer processes often complicates solute transport simulations. We present a new approach for the implementation of the multirate mass transfer model into random walk particle tracking. This novel method allows for a spatially heterogeneous distribution of mass transfer coefficients as well as hydrodynamic parameters in three dimensions, and it is well suited for avoiding numerical dispersion and solving computationally demanding transport simulations. For this purpose the normalized zeroth spatial moments of the multirate transport equations are derived and used as phase transition probabilities. Performing a simple Bernoulli trial on the appropriate phase transition probabilities the particle distribution between the mobile domain and any immobile domain can be determined. The approach is compared satisfactorily to analytical and semianalytical solutions for one-dimensional, advectivedispersive transport with different types of mass transfer. Aspects of the numerical implementation of this approach are presented, and it is demonstrated that two restrictive criteria for the time step size have to be considered. By adjusting the time step size for each grid cell on the basis of the cell specific velocity field and mass transfer rate a correct simulation of solute transport is assured, while at the same time computational efficiency is preserved. Finally, an example is presented evaluating the effect of a heterogeneous intraparticle pore diffusion in a synthetic aquifer. The results demonstrate that for this specific case the heterogeneous distribution of mass transfer rates does not have a significant influence on mean solute transport behavior but that at low concentration ranges, differences between the different mass transfer models become visible.

Ocean Science, 2007

Random walk models are a powerful tool for the investigation of transport processes in turbulent flows. However, standard random walk methods are applicable only when the flow velocities and diffusivity are sufficiently smooth functions. In practice there are some regions where the rapid but continuous change in diffusivity may be represented 5 by a discontinuity. The random walk model based on backwardÎto calculus can be used for these problems. This model was proposed by LaBolle et al. (2000). The latter is best suited to the problems under consideration. It is then applied for two test cases with discontinuous diffusivity, highlighting the advantages of this method. 20 ticles the advection-diffusion processes can be described Costa and Ferreira, 2000; Proehl et al., 2005;.

Fickian assumptions are used in deriving the advection-dispersion equation which models the solute transport in porous media. The hydrodynamic dispersion coefficient defined as a result of these assumptions has been found to be scale dependent. developed a stochastic computational model for solute transport in saturated porous media without using Fickian assumptions. The model consists of two main parameters; correlation length and variance, and the velocity of solute was assumed as a fundamental stochastic variable. In this paper, the stochastic model was investigated to understand its behaviour. As the statistical nature of the model changes with the parameters, the computational solution of the model was explored in relation to the parameters. The variance is found to be the dominant parameter, however, there is a correlation between two parameters and they influence the stochasticity of the flow in a complex manner. We hypothesised that the variance is inversely proportional to the pore size and the correlation length represents the geometry of flow. The computational results of different scales show that the hypotheses are reasonable. The model illustrates that it could capture the scale dependence of dispersivity and mimic the advection-dispersion equation in more deterministic situations.

Water Pollution VIII: Modelling, Monitoring and Management, 2006

Stochastic differential equations (SDEs) are stochastic in nature. The SDEs under consideration are often called particle models (PMs). PMs in this article model the simulation of transport of pollutants in shallow waters. The main focus is the derivation and efficient implementation of an adaptive scheme for numerical integration of the SDEs in this article. The error determination at each integration time step near the boundary where the diffusion is dominant is done by a pair of numerical schemes with strong order 1 of convergence and that of strong order 1.5. When the deterministic is dominant we use the aforementioned order 1 scheme and another scheme of strong order 2. An optimal stepsize for a given error tolerance is estimated. Moreover, the algorithm is developed in such a way that it allows for a completely flexible change of the time stepsize while guaranteeing correct Brownian paths. The software implementation uses the MPI library and allows for parallel processing. By making use of internal synchronisation points it allows for snapshots and particle counts to be made at given times, despite the inherent asynchronicity of the particles with regard to time.

Chemical Engineering Journal, 2000

Molecular collisions with very small particles induce Brownian motion. Consequently, such particles exhibit classical diffusion during their sedimentation. However, identical particles too large to be affected by Brownian motion also change their relative positions. This phenomenon is called hydrodynamic diffusion. Long before this term was coined, the variability of individual particle trajectories had been recognized and a stochastic model had been formulated. In general, stochastic and diffusion approaches are formally equivalent. The convective and diffusive terms in a diffusion equation correspond formally to the drift and diffusion terms of a Fokker-Planck equation (FPE). This FPE can be cast in the form of a stochastic differential equation (SDE) that is much easier to solve numerically. The solution of the associated SDE, via a large number of stochastic paths, yields the solution of the original equation. The three-parameter Markov model, formulated a decade before hydrodynamic diffusion became fashionable, describes one-dimensional sedimentation as a simple SDE for the velocity process {V(t)}. It predicts correctly that the steady-state distribution of particle velocities is Gaussian and that the autocorrelation of velocities decays exponentially. The corresponding position process {X(t)} is not Markov, but the bivariate process {X(t), V (t)} is both Gaussian and Markov. The SDE pair yields continuous velocities and sample paths. The other approach does not use the diffusion process corresponding to the FPE for the three-parameter model; rather, it uses an analogy to Fickian diffusion of molecules. By focusing on velocity rather than position, the stochastic model has several advantages. It subsumes Kynch's theory as a first approximation, but corresponds to the reality that particle velocities are, in fact, continuous. It also profits from powerful theorems about stochastic processes in general and Markov processes in particular. It allows transient phenomena to be modeled by using parameters determined from the steady-state. It is very simple and efficient to simulate, but the three parameters must be determined experimentally or computationally. Relevant data are still sparse, but recent experimental and computational work is beginning to determine values of the three parameters and even the additional two parameters needed to simulate three-dimensional motion. If the dependence of the parameters on solids concentration is known, this model can simulate the sedimentation of the entire slurry, including the packed bed and the slurry-supernate interface. Simulations using half a million particles are already feasible with a desktop computer.

Computational and Applied Mathematics, 2014

The modelling of diffusive terms in particle methods is a delicate matter and several models were proposed in the literature to take such terms into account. The diffusion velocity method (DVM), originally designed for the diffusion of passive scalars, turns diffusive terms into convective ones by expressing them as a divergence involving a so-called diffusion velocity. In this paper, DVM is extended to the diffusion of vectorial quantities in the three-dimensional Navier-Stokes equations, in their incompressible, velocity-vorticity formulation. The integration of a large eddy simulation (LES) turbulence model is investigated and a DVM general formulation is proposed. Either with or without LES, a novel expression of the diffusion velocity is derived, which makes it easier to approximate and which highlights the analogy with the original formulation for scalar transport. From this statement, DVM is then analysed in one dimension, both analytically and numerically on test cases to point out its good behaviour.

Water Resources Research, 1998

Local-scale spatial averaging of pore-scale advection-diffusion equations in porous media leads to advection-dispersion equations (ADEs). While often used to describe subsurface transport, ADEs may pose special problems in the context of diffusion theory. Standard diffusion theory applies only when characteristic coefficients, velocity, porosity, and dispersion tensor, are smooth functions of space. Subsurface porous-material properties, however, naturally exhibit spatial variability. Transitions between material types are often abrupt rather than smooth, such as sand in contact with clay. In such composite porous media, characteristic coefficients in spatially averaged transport equations may be discontinuous. Although commonly called on to model transport in these cases, standard diffusion theory does not apply. Herein we develop diffusion theory for ADEs of transport in porous media. Derivation of ADEs from probabilistic assumptions yields (1) necessary conditions for convergence of diffusion processes to ADEs, even when coefficients are discontinuous, and (2) general probabilistic definitions of physical quantities, velocity, and dispersion tensor. As examples of how the new theory can be applied to theoretical and numerical problems of transport in porous media, we evaluate several random walk methods that have appeared in the water resources literature.

2002

Fickian assumptions are used in deriving the advection-dispersion equation which models the solute transport in porous media. The hydrodynamic dispersion coefficient defined as a result of these assumptions has been found to be scale dependent. Kulasiri and Verwoerd [1999] developed a stochastic computational model for solute transport in saturated porous media without using Fickian assumptions. The model consists of two main parameters; correlation length and variance, and the velocity of solute was assumed as a fundamental stochastic variable. In this paper, the stochastic model was investigated to understand its behaviour. As the statistical nature of the model changes with the parameters, the computational solution of the model was explored in relation to the parameters. The variance is found to be the dominant parameter, however, there is a cOlTelation between two parameters and they influence the stochasticity of the flow in a complex manner. We hypothesised that the variance ...

Mathematical and Computer Modelling, 2009

This paper deals with the simulation of transport of pollutants in shallow water using random walk models and develops several computation techniques to speed up the numerical integration of the stochastic differential equations (SDEs). This is achieved by using both random time stepping and parallel processing.

2003

2002 Mathematics Subject Classification: 65C05.The actual transport of the air pollutants is due to the wind. This normally called “advection of the air pollutants”. Diffusion and deposition are other two major physical processes, which take place during the transport of pollutants in the atmosphere. In this paper we study two classes of grid-free Monte Carlo (MC) algorithms for solving an elliptic boundary value problem, where the partial differential equation contains advection, diffusion and deposition parts. The grid-free MC approach to solve the above equation uses a local integral representation and leads to a stochastic process called a random “Walk on balls” (WOB). In the first class of algorithms, the choice of a transition density function in the Markov chain depends on the radius of the maximal ball, lying inside the domain, in which the problem is defined, and on the parameters of the differential operator. While the choice of a transition density function in the second ...