Anomalous spreading of a density front from an infinite continuous source in a concentration-dependent lattice gas automaton diffusion model

Abstract

A two-dimensional lattice gas automaton (LGA) is used for simulating concentration-dependent diffusion in a microscopically random heterogeneous structure. The heterogeneous medium is initialized at a low density ρ 0 and then submitted to a steep concentration gradient by continuous injection of particles at a concentration ρ 1 > ρ 0 from a one-dimensional source to model spreading of a density front. Whereas the nonlinear diffusion equation generally used to describe concentration-dependent diffusion processes predicts a scaling law of the type φ = xt −1/2 in one dimension, the spreading process is shown to deviate from the expected t 1/2 scaling. The time exponent is found to be larger than 1 2 , i.e. diffusion of the density front is enhanced with respect to standard Fickian diffusion. It is also established that the anomalous time exponent decreases as time elapses: anomalous spreading is thus not a timescaling process.We demonstrate that occurrence of anomalous spreading results from the diffusivity gradient (dD(ρ)/dρ) existing in the concentration-dependent LGA diffusion model. Standard Fickian diffusion appears as a special case which only occurs when (dD(ρ)/dρ) ≈ 0. Decrease of the anomalous exponent with time may indicate that anomalous diffusion is only transient. In any case, the LGA system possesses a very long transitory regime and spreading remains an anomalous superdiffusive process over large period of time. A simple qualitative model, based on the supply and demand principle, is proposed to account for anomalous spreading. A correspondence is finally established between LGA simulations and experimental measurements of one-dimensional water absorption in non-saturated porous materials in which evidence of anomalous spreading was recently reported.

Spreading of a density front in the Küntz–Lavallée model of porous media

“…This finding, however, leads to a paradox: how addition of scatterers can enhance transport of particles? According to Küntz and Lavallée [13], this paradox can be explained by two hypotheses. The first one attributes superdiffusion of particles to a very strong dependence of their diffusivity on concentration.…”

“…Thus, on the one hand, randomly distributed specular scatterers have a very limited impact on the diffusivity along the "x" axis. On the other hand, the mean-free path, and hence the diffusion coefficient in the original, scatterer-free FHP model diverge to infinity as the reduced particle concentration goes to 0 or 1 [13]. Combination of these two effects explains the fundamental property of the KL model: extremely strong dependence of the diffusion coefficient on concentration [13].…”

“…Several features of the model deserve closer attention. First, the model employs specular scatterers to mimic a porous medium as well as to reduce hydrodynamic effects [13]. However, the effect of such scatterers on the diffusion along the "x" axis is very peculiar: the "x" component of the momentum is changed only for those particles, whose velocity vector is parallel to the "x" axis.…”

“…In this paper we focus on the Küntz and Lavallée (KL) model and propose a much simpler interpretation of the simulation results obtained in [13,14]. Our approach indicates that the diffusion in the KL model is normal.…”

Nonclassical Transport Processes in Geologic Media: Review of Field and Laboratory Observations and Basic Physical Concepts

“…The nature of the observed anomalous flow behavior is not clear. Theoretical studies based on the assumption of explicit time dependence of the diffusivity (e.g., Lockington and Parlange, 2003), as well as numerical models (e.g., Küntz and Lavallée, 2003), have shown good agreement with the data but were mostly phenomenological and did not provide any physical explanation. It is not clear from the experiments and models whether the observed transport regimes were mostly transient or asymptotic (i.e., corresponding to large times t ).…”