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

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 ρ0 and then submitted to a steep concentration gradient by continuous injection of particles at a concentration ρ1>ρ0 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 t1/2 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