The infiltration of liquid and the propagation of the moisture front in non-saturated porous media are generally described by a diffusion equation that predicts a scaling law of the type x/(t)1/2 in one dimension. This model, generally referred to as the theory of unsaturated flow, was systematically applied to account for water movements in porous building materials. In this paper, two sets of nuclear magnetic resonance (NMR) one-dimensional water absorption profiles, respectively measured in a fired-clay brick and a limestone specimen, are re-examined according to this model. The reinterpretation of the NMR absorption data provides evidence that the infiltration front does not propagate as t1/2 neither in brick nor in limestone, i.e. the absorption process does not conform to the predictions of the unsaturated flow theory in these materials. A new theoretical model for infiltration, based on the assumption of a non-Fickian diffusion mechanism, is thus introduced. The water transfer in partially saturated materials is assumed to follow the general nonlinear diffusion equation (∂θ/∂t)-(∂/∂x)[D(θ)(∂θ/∂x)n] = 0, with n real. For one-dimensional infiltration, the water content θ can be expressed in terms of the single variable ϕ* = xt-α, with α = 1/(n + 1) and the cumulative water infiltration I is given at any time by I = ∫θ0θ1x dθ = tα∫θ0θ1ϕ* dθ = S*tα. The NMR absorption data are shown to be compatible with a non-Fickian diffusion process scaling as t0.58 in brick and as t0.61 in the limestone specimen. The application of the new anomalous diffusion model to brick indicates that the previous t1/2 relation may underestimate the volume of absorbed water by about 30% after only 100 hours. This result has particular relevance for evaluating the durability of building structures.

A one dimensional lattice gas model is used to study the interaction of fluid flows with solid houndaries. Various interaction mechanisms are examined. Lattice Boltzmann simulations show that bounce-back reflection is not the only interaction that yields "no-slip"boundary conditions (zéro velocity at a fixed wall) and that Knudsen type interaction is also appropnate.

In this note, we re-examine one-dimensional diffusion experiments of high concentration aqueous CuSO4 into deionized water carried out by Carey A E et al (1995 Water Resources Res. 31 2213–18). We show that spreading of concentrated CuSO4 fronts does not conform to the t1/2 relation expected from Fick's hypothesis but is subdiffusive; i.e. the cumulated concentration of CuSO4 increases much slower than t1/2. Occurrence of subdiffusion of CuSO4 fronts can be related to the fact that the diffusion coefficient, D(C), of aqueous CuSO4 decreases with increasing C, as previously predicted in Küntz M and Lavallée P (2003 J. Phys. D: Appl. Phys. 36 1135–42). This result supports our initial assumption that non-Fickian diffusion is the rule in all concentration-dependent diffusivity processes, i.e. in diffusion processes involving variation of the diffusion coefficient, D(C), with the concentration of the transported quantity, C. A simple ‘offer and demand’ model is proposed that accounts qualitatively well for all the available experimental data. Spreading fronts are subdiffusive for D(C) decreasing with C, superdiffusive for increasing D(C) and scale as t1/2 only for constant D.

A 2-D lattice gas is used to calculate the effective electrical conductivity of saturated porous media as a function of porosity and conductivity ratio R c between the pore-filling fluid and the solid matrix for various microscopic structures of the pore space. The way the solid phase is introduced allows the porosity φ to take any value between 0 and 1 and the geometry of the pore structure to be as complex as desired. The results are presented in terms of the formation factor F = σ w /σ r , with σ r the effective conductivity of the saturated rock and σ w the conductivity of the fluid. It is shown that the formation factor F as a function of the porosity φ follows a power law F = aφ −m , equivalent to the empirical Archie's law. The exponent m varies with the microgeometry of the pore space and could therefore reflect the microstructure at the macroscopic scale. The prefactor a of the power law, however, is close to 1 regardless of the microstructure. For a given microgeometry of the pore space, the variation of the residual electrical conductivity of the solid matrix induced by a finite conductivity ratio R c does not significantly influence the variation of the effective conductivity of the fluid-solid binary mixture unless the porosity is low.

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.