When can we assume Lagrangian stationarity?

Water Resources Research, 1996

Recently, an exact Eulerian-Lagrangian theory of advective transport in spacetime random velocity fields was developed by one of us. We present a formal extension of this theory that accounts for anisotropic local dispersion. The resultant (conditional) mean transport equation is generally nonlocal in space-time. To assess the impact of local dispersion on the prediction of transport under uncertainty, we adopt a first-order pseudo-Fickian approximation for this equation. We then solve it numerically by Galerkin finite elements for two-dimensional transport from an instantaneous square source in a uniform (unconditional) mean flow field subject to isotropic local dispersion. We use a higherorder approximation to compute explicitly the standard deviation and coefficient of variation of the predicted concentrations. Our theory shows (in an exact manner), and our numerical results demonstrate (under the above closure approximations), that the effect of local dispersion on first and second concentration moments varies monotonically with the magnitude of the local dispersion coefficient. When this coefficient is small relative to macrodispersion, its effect on the prediction of nonreactive transport under uncertainty can, for all practical purposes, be disregarded. This is contrary to some recent assertions in the literature that local dispersion must always be taken into account, no matter how small.

Journal of Physics A: Mathematical …, 2009

2020

In this paper we extend the homogenization results obtained in (G. Allaire, A. Mikelic, A. Piatnitski, J. Math. Phys. 51 (2010), 123103) for a system of partial differential equations describing the transport of a N-component electrolyte in a dilute Newtonian solvent through a rigid periodic porous medium, to the case of random disperse porous media. We present a study of the nonlinear Poisson-Boltzmann equation in a random medium, establish convergence of the stochastic homogenization procedure and prove well-posedness of the two-scale homogenized equations. In addition, after separating scales, we prove that the effective tensor satisfies the so-called Onsager properties, that is the tensor is symmetric and positive definite. This result shows that the Onsager theory applies to random porous media. The strong convergence of the fluxes is also established.

Physical Review E, 2010

Whenever one uses translation invariant mean Green's functions to describe the behavior in the mean and to estimate dispersion coefficients for diffusion in random velocity fields, the spatial homogeneity of the transition probability of the transport process is implicitly assumed. This property can be proved for deterministic initial conditions if, in addition to the statistical homogeneity of the space-random velocity field, the existence of unique classical solutions of the transport equations is ensured. When uniqueness condition fails and translation invariance of the mean Green's function cannot be assumed, as in the case of nonsmooth samples of random velocity fields with exponential correlations, asymptotic dispersion coefficients can still be estimated within an alternative approach using the Itô equation. Numerical simulations confirm the predicted asymptotic behavior of the coefficients, but they also show their dependence on initial conditions at early times, a signature of inhomogeneous transition probabilities. Such memory effects are even more relevant for random initial conditions, which are a result of the past evolution of the process of diffusion in correlated velocity fields, and they persist indefinitely in case of power law correlations. It was found that the transition probabilities for successive times can be spatially homogeneous only if a long-time normal diffusion limit exits. Moreover, when transition probabilities, for either deterministic or random initial states, are spatially homogeneous, they can be explicitly written as Gaussian distributions.

Physical Review E, 2016

We present a strong relationship between the microstructural characteristics of, and the fluid velocity fields confined to, three-dimensional random porous materials. The relationship is revealed through simultaneously extracting correlation functions R uu (r) of the spatial (Eulerian) velocity fields and microstructural two-point correlation functions S 2 (r) of the random porous heterogeneous materials. This demonstrates that the effective physical transport properties depend on the characteristics of complex pore structure owing to the relationship between R uu (r) and S 2 (r) revealed in this study. Further, the mean excess plot was used to investigate the right tail of the streamwise velocity component that was found to obey light-tail distributions. Based on the mean excess plot, a generalized Pareto distribution can be used to approximate the positive streamwise velocity distribution.

Physical Review Letters, 2013

We study the intermittency of fluid velocities in porous media and its relation to anomalous dispersion. Lagrangian velocities measured at equidistant points along streamlines are shown to form a spatial Markov process. As a consequence of this remarkable property, the dispersion of fluid particles can be described by a continuous time random walk with correlated temporal increments. This new dynamical picture of intermittency provides a direct link between the microscale flow, its intermittent properties, and non-Fickian dispersion.

Water Resources Research, 1981

Stochastic solutions of the differential equation describing one-dimensional flow through a porous medium with spatially variable hydraulic conductivity which is represented by a stationary (statistically homogeneous) random process are developed using several techniques. The analytical approximations using first-order analysis (propagation of error), covariance differential equations, and variogram analysis all yield consistent results which demonstrate the important effects of boundary conditions and conditioning. Using the theory of intrinsic random functions, stochastic solutions are developed for the case when the logarithm of the hydraulic conductivity is a three-dimensional stationary random field. In contrast to the one-dimensional case, it is found that the resulting thr•e-dimensional head perturbation will be locally stationary under very general conditions. Results from the one-dimensional analytical solutions are found to be in agreement with previous Monte Carlo simulations for a flow system of finite length. The solution based on linearization in the logarithm of hydraulic conductivity proved to be very robust, showing reasonable agreement with Monte Carlo results even for the largest input standard deviation of ouar --4.6.

2020

• Development of a new approach to reactive transport modeling that accounts for the 8 dynamics of small-scale concentration fluctuations 9 • The temporal evolution of integrated mixing metrics agrees with the characteristic trends of fully-resolved systems • Experimental observations of mixing-limited reactive transport are successfully reproduced

Water Resources Research, 2009

1] In this paper we present flow and travel time ensemble statistics based on a new simulation methodology, the adaptive Fup Monte Carlo method (AFMCM). As a benchmark case, we considered two-dimensional steady flow in a rectangular domain characterized by multi-Gaussian heterogeneity structure with an isotropic exponential correlation and lnK variance s Y 2 up to 8. Advective transport is investigated using the travel time framework where Lagrangian variables (e.g., velocity, transverse displacement, or travel time) depend on space rather than on time. We find that Eulerian and Lagrangian velocity distributions diverge for increasing lnK variance due to enhanced channeling. Transverse displacement is a nonnormal for all s Y 2 and control planes close to the injection area, but after xI Y = 20 was found to be nearly normal even for high s Y 2 . Travel time distribution deviates from the Fickian model for large lnK variance and exhibits increasing skewness and a power law tail for large lnK variance, the slope of which decreases for increasing distance from the source; no anomalous features are found. Second moment of advective transport is analyzed with respect to the covariance of two Lagrangian velocity variables: slowness and slope which are directly related to the travel time and transverse displacement variance, which are subsequently related to the longitudinal and transverse dispersion. We provide simple estimators for the Eulerian velocity variance, travel time variance, slowness, and longitudinal dispersivity as a practical contribution of this analysis. Both two-parameter models considered (the advection-dispersion equation and the lognormal model) provide relatively poor representations of the initial part of the travel time probability density function in highly heterogeneous porous media. We identify the need for further theoretical and experimental scrutiny of early arrival times, and the need for computing higher-order moments for a more accurate characterization of the travel time probability density function. A brief discussion is presented on the challenges and extensions for which AFMCM is suggested as a suitable approach.

2014

We investigate the velocity distribution function of the fluid flow through a model of random porous media. We examine how the form of the distribution changes with porosity. We find a crossover porosity at which the scaling character of the distribution changes and discuss our findings in the context of a nearly exponential velocity distribution function measured recently experimentally [

Computational Geosciences, 2006

We consider groundwater steady flow in a heterogeneous porous formation of random and stationary log-conductivity Y = ln K, characterized by the mean bY À, and the two point correlation function C Y which in turn has finite, and different horizontal and vertical integral scales I and I v , respectively. The fluid velocity V, driven by a given head drop applied at the boundary, has constant mean value U K (U, 0, 0). Approximate explicit analytical expressions for transverse velocity covariances are derived. The adopted methodology follows the approach developed by Dagan and Cvetkovic (Spatial moments of kinetically sorbing plume in a heterogeneous aquifers, Water Resour. Res. 29 (1993) 4053) to obtain a similar result for the longitudinal velocity covariance. Indeed, the approximate covariances of transverse velocities are determined by requiring that they have the exact first order variances as well as zero integral scale (G. Dagan, Flow and Transport in Porous Formations (Springer, 1989)) , and provide the exact asymptotic limits of the displacement covariance of the fluid particles obtained by Russo (On the velocity covariance and transport modeling in heterogeneous anisotropic porous formations 1. Saturated flow, Water Resour. Res., 31 (1995) 129). Comparisons with numerical results show that the proposed expressions compare quite well in the early time regime, and for Ut/I >100. Since most of the applications, like assessing the effective mobility of contaminants or quantifying the potential hazards of nuclear repositories, require predictions over higher times the proposed approximate expressions provide acceptable results. The main advantage related to such expressions is that they allow obtaining closed analytical forms of spatial moments pertaining to kinetically sorbing contaminant plumes avoiding the very heavy computational effort which is generally demanded. For illustration purposes, we consider the movement of one contaminant species, and show how our approximate spatial moments compare with the numerical simulations.

Journal of Hydrology, 2003

In this study, we make use of a nonstationary stochastic theory in studying solute flux through spatially nonstationary flows in porous media. The nonstationarity of flow stems from various sources, such as multi-scale, nonstationary medium features and complex hydraulic boundary conditions. These flow nonstationarities are beyond the applicable range of the 'classical' stochastic theory for stationary flow fields, but widely exist in natural media. In this study, the stochastic frames for flow and transport are developed through an analytical analysis while the solutions are obtained with a numerical method. This approach combines the stochastic concept with the flexibility of the numerical method in handling medium nonstationarity and boundary/initial conditions. It provides a practical way for applying stochastic theory to solute transport in complex groundwater environments. This approach is demonstrated through some synthetic cases of solute transport in multi-scale media as well as some hypothetical scenarios of solute transport in the groundwater below the Yucca Mountain project area. It is shown that the spatial variations of mean log-conductivity and correlation function significantly affect the mean and variance of solute flux. Even for a stationary medium, complex hydraulic boundary conditions may result in a nonstationary flow field. Flow nonstationarity and/or nonuniform distribution of initial plume (geometry and/or density) may lead to nonGaussian behaviors (with multiple peaks) for mean and variance of the solute flux. The calculated standard deviation of solute flux is generally larger than its mean value, which implies that real solute fluxes may significantly deviate from the mean predictions. q

Journal of Hydrology, 2004

Journal of Fluid Mechanics, 2003

We consider flow in partially saturated heterogeneous porous media with uncertain hydraulic parameters. By treating the saturated conductivity of the medium as a random field, we derive a set of deterministic equations for the statistics (ensemble mean and variance) of fluid pressure. This is done for three constitutive models that describe the nonlinear dependence of relative conductivity on pressure. We use the Kirchhoff transform to map Richards equation into a linear PDE and explore alternative closures for the resulting moment equations. Regardless of the type of nonlinearity, closure by perturbation is more accurate than closure based on the non-perturbative Gaussian mapping. We also demonstrate that predictability of unsaturated flow in heterogeneous porous media is enhanced by choosing either the Brooks-Corey or van Genuchten constitutive model over the Gardner model.

Water Resources Research, 2010

1] The travel time formulation of advective transport in heterogeneous porous media is of interest both conceptually, e.g., for incorporating retention processes, and in applications where typically the travel time peak, early, and late arrivals of contaminants are of major concern in a regulatory or remediation context. Furthermore, the travel time moments are of interest for quantifying uncertainty in advective transport of tracers released from point sources in heterogeneous aquifers. In view of this interest, the travel time distribution has been studied in the literature; however, the link to the hydraulic conductivity statistics has been typically restricted to the first two moments. Here we investigate the influence of higher travel time moments on the travel time probability density function (pdf) in heterogeneous porous media combining Monte Carlo simulations with the maximum entropy principle. The Monte Carlo experimental pdf is obtained by the adaptive Fup Monte Carlo method (AFMCM) for advective transport characterized by a multi-Gaussian structure with exponential covariance considering two injection modes (in-flux and resident) and lnK variance up to 8. A maximum entropy (MaxEnt) algorithm based on Fup basis functions is used for the complete characterization of the travel time pdf. All travel time moments become linear with distance. Initial nonlinearity is found mainly for the resident injection mode, which exhibits a strong nonlinearity within first 5I Y for high heterogeneity. For the resident injection mode, the form of variance and all higher moments changes from the familiar concave form predicted by the firstorder theory to a convex form; for the in-flux mode, linearity is preserved even for high heterogeneity. The number of moments sufficient for a complete characterization of the travel time pdf mainly depends on the heterogeneity level. Mean and variance completely describe travel time pdf for low and mild heterogeneity, skewness is dominant for lnK variance around 4, while kurtosis and fifth moment are required for lnK variance higher than 4. Including skewness seems sufficient for describing the peak and late arrivals. Linearity of travel time moments enables the prediction of asymptotic behavior of the travel time pdf which in the limit converges to a symmetric distribution and Fickian transport. However, higher-order travel time moments may be important for most practical purposes and in particular for advective transport in highly heterogeneous porous media for a long distance from the source.