An Analytical Model of Fickian and Non-Fickian Dispersion in Evolving-Scale Log-Conductivity Distributions

Abstract

Abstract:The characteristics of solute transport within log-conductivity fields represented by power-law semi-variograms are investigated by an analytical Lagrangian approach that accounts for the automatic frequency cut-off induced by the initial contaminant plume size. The transport process anomaly is critically controlled by the magnitude of the Péclet number. Interestingly enough, unlike the case of fast-decaying correlation functions (i.e., exponential or Gaussian), the presence of intensive transverse diffusion acts as an antagonist mechanism in the process of Fickian regime achievement. On the other hand, for markedly advective conditions and finite initial plume size, even the ergodic longitudinal dispersion coefficient turns out to be asymptotically constant, and the corresponding expected concentration distribution can therefore be obtained by conventional mathematical methods.

A Theoretical Study about Ergodicity Issues in Predicting Contaminant Plume Evolution in Aquifers

“…Following the standard linearized transport formulation (e.g., [2]), the velocity field sampling trajectory consists of a mean-velocity straight line (a + Vt), with a indicating the generic particle starting position, only perturbed by the local dispersive Brownian displacement (X B ). In these conditions: (13) where Ω 0 indicates the injection volume, δ the Dirac Delta, and C 0 (a) the initial concentration distribution. As mentioned when introducing Equation 1, in the present study it is assumed that the initial concentration distribution is a point-like one: C 0 (a) → δ(a) .…”

“…In terms of position and dimensions, the discussion is based on centroid and central inertia moments, as well as on the conditions under which they can be considered as equivalent to same-order single-particle statistical moments; in terms of point concentrations, the discussion is based on concentration ensemble mean, variance and coefficient of variation (which in turn involve the single-particle moments as well as the statistics of the barycenter of mass) and the conditions under which the Gaussian ensemble mean for point instantaneous mass injection can be representative of the actual distribution. The mathematical treatment, which also hinges on previous author's results, makes use of both Lagrangian (e.g., [8][9][10][11][12][13]) and Eulerian (e.g., [14][15][16]) framework for tracer transport in heterogeneous flow fields.…”

Ensemble and effective dispersion in three-dimensional isotropic fractal media

“…9such that the singularity at jxj ¼ 0 is avoided, whereas the behavior for large distances with jxj=l ) 1 is similar to that of a purely fractal medium as used by e.g. Dagan (1994), Bellin et al (1996), Fiori (2001) and Pannone (2017). Real subsurface formations are always connected to some physical length scales, be it the total size of the aquifer or the support scale induced by the measurement process or the modeling assumptions (Neuman 1994;Neuman and Tartakovsky 2009).…”

“…This surprising similarity between fractal and classical media was due to the aforementioned truncation of the fractal behavior at larger scales, which made their variograms (see Di Federico and Neuman 1998b) as well as the resulting transport behavior very alike. More recently Pannone (2017) investigated the impact of the Péclet number of the longitudinal macrodispersion coefficient, showing that both Fickian and non-Fickian behavior can be found depending on the fractality of the medium. Despite this large number of studies, there is no study to this date, where longitudinal and transverse dispersion are jointly investigated in a fully three-dimensional medium and explicit descriptions for the dispersion coefficients are presented.…”

Editorial of Special Issue “Advances in Groundwater Flow and Solute Transport: Pushing the Hidden Boundary”

“…For this subject, Pannone [1] has adopted a stochastic approach to deal with an evolving-scale heterogeneous formation using power-law semi-variograms. Pannone [1] has analytically shown that dispersion in such a hierarchy system can be ergodic and Fickian or non-ergodic and super-diffusive, based on the scaling exponent value and the magnitude of Peclet number, which was defined in this study as the ratio of the product of the ensemble mean velocity at the initial plume size to the local dispersion. Specifically, a large Peclet number will make the transport process closer to asymptotically ergodic-Fickian conditions.…”