Experimental evidence and theoretical analysis of anomalous diffusion during water infiltration in porous building materials

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 t^1/2 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 = ∫_θ0^θ₁x dθ = t^α∫_θ0^θ₁ϕ* dθ = S*t^α. The NMR absorption data are shown to be compatible with a non-Fickian diffusion process scaling as t^0.58 in brick and as t^0.61 in the limestone specimen. The application of the new anomalous diffusion model to brick indicates that the previous t^1/2 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.