An effective scale-dependent dispersivity deduced from a purely convective flow field

Abstract

In the case of straight flow but with hydraulic conductivity varying in a transverse direction, the distribution of hydraulic conductivity has been determined for which the breakthrough curve due to convection only will have the same analytical form as the onedimensional convection/dispersion equation solution at the outlet end of a porous medium. That distribution is found exactly and it is very similar to the lognormal distribution. This result is significant since field evidence indicates that the logarithm of hydraulic conductivity is normally distributed. For the case considered, a simple relation between dispersivity and the coefficient of variation of hydraulic conductivity is found. One can thus determine very simply dispersivity in terms of the parameters of the distribution of hydraulic conductivity. This is particularly useful to estimate dispersivity in various cells of finite difference or finite element models when the distribution of hydraulic conductivity is not stationary, i.e. varies in space. Dérivation d'une dispersivité équivalente, fonction de l'échelle, à partir d'un champ d'écoulement purement convectifRésumé Dans le cas d'un écoulement droit mais avec variation de conductivité hydraulique dans la direction transversale, on a obtenu la distribution pour laquelle l'écoulement purement convectif et l'écoule-ment convectif/diffusif dans une seule dimension donnent la même courbe de concentration avec le temps à la sortie du milieu poreux. Cette distribution théorétique se confond presque avec la loi normale logarithmique, qui a été retenue pour caractériser la variation aléatoire de la conductivité sur le terrain. Le résultat est donc pratique et significatif. On a de plus obtenu une relation simple entre la dispersivité et le coefficient de variation de la conductivité. On peut ainsi déterminer de façon très simple la dispersives en termes de paramètres de la distribution de la conductivité hydraulique. Cette relation permet d'estimer différentes dispersivités dans les mailles d'un réseau de différences finies quand la distribution de la conductivité n'est pas stationnaire dans l'espace, ni dans sa moyenne ni dans sa variance.

A parsimonious approach for large-scale tracer test interpretation

“…This is equivalent to the concept of heterogeneous advection used by Becker and Shapiro (2003) to describe mass spreading related to separation of advective pathways (Berkowitz et al 2006). The scaling of dispersive properties has also been interpreted as a consequence of the use of a n-1 dimension analytical solution to a n-dimension problem (Pickens andGrisak 1981)-Morel-Seytoux andNachabe (1992) demonstrated that for permanent flow conditions with pure advection in 2D that an equivalent one-dimensional (1D) macro-dispersivity transport scheme can be used with a dispersivity that will be a linear function of the scale. This work introduces a proportionality factor between macroscopic dispersivity and scale, which was also used by Pickens and Grisak (1981) for well-to-well tracer tests performed in a sandy stratified aquifer.…”

Dynamics of groundwater mounds: analytical solutions and integral characteristics

“…Simple analytical solutions for transient regimes (of plug or periodic type) with explicit expressions for the Hamiltonian (stream function \\i (x, y, t)) of the system of order 3/2 can serve as test procedures for more realistic models involving dissipation (compressible matrix or fluid). The phenomenon of dispersion is usually related with aquifer inhomogeneity and parameters of the advective dispersion equation are connected with purely convective flow and conductivity distributions (Morel-Seytoux & Nachabe, 1992). Meanwhile transient fluctuations of a seepage flow seem to be able produce dispersion (or chaos) even in homogeneous porous media.…”