Itˆoitˆ Itˆo Equation Model for Dispersion of Solutes in Heterogeneous Media

For transport in statistically homogeneous random velocity flelds 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 be- havior 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 be- tween initial positions and displacements of solute molecules. Reliable numerical exper- iments show that increasing the source dimensions has two opposite efiects: it reduces the uncertainty related to the randomness of center of mass, but, at the same time, it yields large memory terms. The memory efiects increase with the source dimension and depend on its shape and orientation. Large narrow sources oriented transverse to the mean ∞ow direction yield ergodic beha...

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.

Water Resources Research, 2002

The basic conceptual picture and theoretical basis for development of transport equations in porous media are examined. The general form of the governing equations is derived for conservative chemical transport in heterogeneous geological formations, for single realizations and for ensemble averages of the domain. The application of these transport equations is focused on accounting for the appearance of non-Fickian (anomalous) transport behavior. The general ensemble-averaged transport equation is shown to be equivalent to a continuous time random walk (CTRW) and reduces to the conventional forms of the advection-dispersion equation (ADE) under highly restrictive conditions. Fractional derivative formulations of the transport equations, both temporal and spatial, emerge as special cases of the CTRW. In particular, the use in this context of Lévy flights is critically examined. In order to determine chemical transport in field-scale situations, the CTRW approach is generalized to non-stationary systems. We outline a practical numerical scheme, similar to those used with extended geological models, to account for the often important effects of unresolved heterogeneities. * Electronic address: brian.berkowitz@weizmann.ac.il † Electronic address: klafter@post.tau.ac.il ‡ Electronic address: metz@nordita.dk § Electronic address: harvey.scher@weizmann.ac.il

Physica A: Statistical Mechanics and its Applications, 1998

We study the statistics of relative distances R(t) between uid particles in a spatially smooth random ow with arbitrary temporal correlations. Using the space dimensionality d as a large parameter we develop an e ective description of Lagrangian dispersion. We describe the exponential growth of relative distances R 2 (t) ∝ exp 2 t at di erent values of the ratio between the correlation and turnover times. We ÿnd the stretching correlation time which determines the dependence of R 1R2 on the di erence t1 − t2. The calculation of the next cumulant of R 2 shows that statistics of R 2 is nearly Gaussian at small times (as long as d1) and becomes log-normal at large times when large-d approach fails for high-order moments. The crossover time between the regimes is the stretching correlation time which surprisingly appears to depend on the details of the velocity statistics at t. We establish the dispersion of the ln(R 2) in the log-normal statistics.

Combustion and Flame, 1998

The prediction of particle dispersion by interactions with a turbulent gaseous fluid is an important, yet difficult, problem. This paper presents a new model to predict the motion of particles in a turbulent flow. This model, which solves for the probability density function (pdf) for particle velocity, treats the impact of the turbulence on the velocity pdf as a diffusion process. Particle concentrations are, in turn, found from the velocity distributions. Comparisons between the model predictions and both analytical and experimental results are presented. Results are reported for flows of homogeneous, isotropic turbulence; for grid-generated turbulence; and for round jets. This study includes a large range of particle diameters and densities. Good agreement is found between the predictions and measurements.

2008

Memory effects in the pre-asymptotic regime of transport in heterogeneous media, indicated by reliable numerical experiments, are explained by a dependence on initial conditions which is specific for systems with space variable properties such as aquifers, turbulent atmosphere or ionized plasmas. For statistically homogeneous random velocity fields, with finite correlation range, incompressible, and continuously differentiable, the one-particle dispersion (average over velocity realizations of the actual dispersion for fixed initial positions of the solute molecules) is a "memory-free" quantity, solely determined by the velocity correlation and the local dispersion coefficient and 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. 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 ergodically. Such memory effects are significantly large over hundreds of heterogeneity scales and should therefore be considered in practical applications, as for instance calibration of model parameters, forecasting, identification of the contaminant source.

Mathematical geology, 1999

Dispersive mass transport processes in naturally heterogeneous geological formations (porous media) are investigated based on a particle approach to mass transport and on its numerical implementation using LPT3D, a Lagrangian Particle Tracking 3D code. We are currently using this approach for studying microscale and macroscale space-time behaviour (advection, diffusion, dispersion) of tracer plumes, solutes or miscible fluids, in 1,2,3-dimensional heterogeneous and anisotropic subsurface formations (aquifers, petroleum reservoirs). Our analyses are based on a general advection-diffusion model and numerical scheme where concentrations and fluxes are discretized in terms of particles. The advection-diffusion theory is presented in a probabilistic framework, and in particular, a numerical analysis is developed for the case of advective transport and rotational flows (numerical stability of the explicit Euler scheme). The remainder of the paper is devoted to the behaviour of concentration, mass flux density, and statistical moments of the transported tracer plume in the case of heterogeneous steady flow fields, where macroscale dispersion occurs due to geologic heterogeneity and stratification. We focus on the case of perfectly stratified or multilayered media, obtained by generating many horizontal layers with a purely random transverse distribution of permeability and horizontal velocity. In this case, we calculate explicitly the exact mass concentration field C(x,t), mass flux density field f(x,t), and moments. This includes spatial moments and dispersion variance σ 2 X (t) on a finite domain L, and temporal moments on a finite time scale T, e.g. the "mass variance" of arrival times σ T 2 (x). The moments are related to flux-concentrations in a way that takes explicitly into account finite spacetime scales of analyses (time-dependent tracer mass; spatially variable "flow through" mass). The multilayered model problem is then used in numerical experiments for testing different ways of recovering information on tracer plume migration, dispersion, concentration and flux fields. Our analyses rely on a probabilistic interpretation that emerges naturally from the particle approach; it is based on spatial moments (particle positions), temporal moments (mass weighted arrival times), and probability densities (both concentrations and fluxes). Finally, as an alternative to direct estimations of the flux and concentration fields, we formulate and study the "Moment Inverse Problem". Solving the M.I.P yields an indirect method for estimating the space-time distribution of flux-concentrations based on observed or estimated moments of the plume. The moments may be estimated from field measurements, or numerically computed by particle tracking as we do here.

Water Resources Research, 2008

1] 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 ergodically. Such memory effects are significantly large over hundreds of heterogeneity scales and should therefore be considered in practical applications, as for instance calibration of model parameters, forecasting, identification of the contaminant source.

2010

Derivations of continuum nonlocal models of non-Fickian (anomalous) transport require assumptions that might limit their applicability. We present a particle-based algorithm, which obviates the need for many of these assumptions by allowing stochastic processes that represent spatial and temporal random increments to be correlated in space and time, be stationary or non-stationary, and to have arbitrary distributions. The approach treats a particle trajectory as a subordinated stochastic process that is described by a set of Langevin equations, which represent a continuous time random walk (CTRW). Convolutionbased particle tracking (CBPT) is used to increase the computational efficiency and accuracy of these particle-based simulations. The combined CTRW-CBPT approach enables one to convert any particle tracking legacy code into a simulator capable of handling non-Fickian transport.

Newcastle University eBooks, 2016