Harald Kinzelbach

Temporal behavior of a solute cloud in a heterogeneous porous medium: 1. Point‐like injection

We investigate the temporal behavior of transport coefficients in a model for transport of a solute through a spatially heterogeneous saturated aquifer. In the framework of a stochastic approach we derive explicit expressions for the temporal behavior of the center-of-mass velocity and the dispersion of the concentration distribution after a point-like injection of solute at time t=0, using a second-order perturbation expansion. The model takes into account local variations in the hydraulic conductivity (which, in turn, induce local fluctuations in the groundwater flow velocities) and in the chemical adsorption properties of the medium (which lead to a spatially varying local retardation factor). In the given perturbation theory approach the various heterogeneity-induced contributions can be systematically traced back to fluctuations in these quantities and to cross correlations between them. We analyze two conceptually different definitions for the resulting dispersion coefficient: the “effective”dispersion coefficient which is derived from the average over the centered second moments of the spatial concentration distributions in every realization and the “ensemble” dispersion coefficient which follows from the second moment of the ensemble-averaged concentration distribution. The first quantity characterizes the dispersion in a typical realization of the medium, whereas the second one describes the (formal) dispersion properties of the ensemble as a whole. We give explicit analytic expressions for both quantities as functions of time and show that for finite times their temporal behavior is remarkably different. The ensemble dispersion coefficient which is usually evaluated in the literature considerably overestimates the dispersion typically found in one given realization of the medium. From our explicit results we identify two relevant timescales separating regimes of qualitatively and quantitatively different temporal behavior: The shorter of the two scales is set by the advective transport of the solute cloud over one disorder correlation length, whereas the second, much larger one, is related to the dispersive spreading over the same distance. Only for times much larger than this second scale, do the effective and the ensemble dispersion coefficient become equivalent because of mixing caused by the local transversal dispersion. The formulae are applied to the Borden experiment data. It is concluded that the observed dispersion coefficient matches the effective dispersion coefficient at finite times proposed in this paper very well.

Temporal behavior of a solute cloud in a heterogeneous porous medium: 2. Spatially extended injection

We investigate the temporal behavior of transport coefficients in a stochastic model for transport of a solute through a spatially heterogeneous saturated aquifer. While the first of these two companion papers [Dentz et al., this issue] investigated a situation characterized by a point-like solute injection, we now focus on the case of spatially extended solute sources. The analysis of the finite time behavior of the transport coefficients makes it necessary to distinguish between two fundamentally different quantities characterizing the solute dispersion. We define an “effective” dispersion coefficient which is derived from the average over the centered second moments of the spatial concentration distributions in every realization and an “ensemble” dispersion coefficient which follows from the second moment of the ensemble-averaged concentration distribution. While the two quantities are equivalent in the asymptotic limit of infinite times or infinitely extended sources, they are qualitatively and quantitatively different for the more realistic situation of finite times and finite source extent. We demonstrate that in this case the ensemble quantity, used more or less implicitly in most of the previous studies, overestimates the true dispersion of the plume. Using a second-order perturbation theory approach, we derive explicit solutions for the temporal behavior of the dispersion coefficients for various types of isotropic and anisotropic initial conditions. We identify the relevant timescales which separate regimes of different temporal behavior and apply our formulae to the Borden experiment data. We find a good agreement between theory and experiment if we compare the observed dispersion with the appropriate effective dispersion coefficient (including the leading effects of the local dispersion), whereas the ensemble dispersion coefficient commonly used in the literature to analyze these data overestimates the experimental results considerably.

Effective parameters in heterogeneous and homogeneous transport models with kinetic sorption

Transport of dissolved tracers undergoing kinetic sorption in saturated porous media is described on the basis of a dual-porosity model with heterogeneous and cross-correlated sorption parameters, i.e., distribution coefficient and exchange rate. The approach is a conceptual model for reactive transport in a medium with spatially varying reaction capacity, given by the distribution coefficient, and spatially varying accessibility, given by the exchange rate. We treat the sorption parameters as a stochastic process and apply a perturbation approach. From the ensemble-averaged spatial moments of a plume, we analytically derive formal expressions for time-dependent effective transport parameters. For vanishing microdispersion the calculations are carried out up to second order for the effective transport velocity ueff(t) and the effective dispersion coefficient Deff(t). For large times the effective retardation is determined by the ensemble-averaged distribution coefficient, whereas the effective dispersion is related to the sorption parameters in a more complicated way depending on the variability of the exchange rate and of the distribution coefficient. Effective sorption parameters are given. For comparison we derive exact expressions for ueff(t) and Deff(t) in a homogeneous triple-porosity model. Unlike the simpler homogeneous dual-porosity model, the triple-porosity model yields a satisfactory description of the time-dependent dispersion of the heterogeneous model. The appropriate sorption parameters for the triple-porosity model are given as functions of the stochastic parameters.

Transients and basins of attraction in neutral network models

Approximate dynamic mean field equations for a generalized Hopfield model are derived, which allow to calculate transient properties of this model. These equations are exact for short times and yield the replica symmetric solution as a stationary solution. They allow reliable computation of retrieval trajectories and basins of attraction of retrieval states, as demonstrated by comparison with simulations. The equations are derived for networks with arbitrary mean activity and results are given for the standard model and for low activities.

EFFECTIVE DISPERSION OF A SOLUTE CLOUD IN A CHEMICALLY HETEROGENEOUS POROUS MEDIUM : COMPARISON OF TWO ENSEMBLE-AVERAGING PROCEDURES

We consider solute transport in a porous medium for which we assume the retardation factor R(x), resulting from linear chemical adsorption, to be stochastically varying in space. For large times, the evolution of a solute plume developing from a pointlike, instantaneous solute injection is described by its effective velocity and dispersion. We calculate such quantities using perturbation theory and two different averaging procedures. The first and correct procedure calculates the central moments of the cloud for a given aquifer realization and averages over the ensemble afterward. The second method, which is mathematically less demanding, obtains large-scale transport coefficients from the moments of the ensemble-averaged concentration distribution. This last approach is often used to replace the correct procedure, tacitly assuming that the two averaging methods lead to the same effective quantities. We show that the results actually differ in one dimension, whereas the difference vanishes in higher dimensions. The effective retardation factor is found to be the ensemble average of the corresponding small-scale quantity. The effective dispersion coefficient, on the other hand, differs from the retarded small-scale dispersion coefficient. It is significantly enhanced by the inhomogeneous fluctuations of the disordered medium.

Dynamics of the finite SK spin-glass

The dynamical behavior of a Sherrington-Kirkpatrick spin-glass model consisting of a large but finite number of Ising spins with a time evolution given by Glauber dynamics is investigated. Starting from the resummation of a diagrammatic expansion we derive a differential equation for the response function which allows us to handle nonperturbative effects. This enables us to find explicit expressions for the dynamical behavior of response and correlation function on time scales related to those free energy barriers which diverge with system sizeN. For the largest of these barriers we find a behavior proportional toNλ with λ=1/3.