Anomalous infiltration in partially saturated porous media

Many critical environmental problems involve fluid infiltration into an initially dry porous medium and in such realistic settings the infiltration front is a partially saturated fringe and the medium is heterogeneous. Here we consider the case of horizontal infiltration into two-component layered media where a background of high saturated hydraulic conductivity contains inclusions of much lower conductivity laid out as Cantor set patterns. The process is modeled by Richards equation of partially saturated horizontal flow with appropriate constitutive models relating pressure head, moisture, and unsaturated conductivity. Numerical simulations with high contrasts of saturated conductivities show that an infiltration length scales in time with superdiffusive (>1/2) exponents that depend on the fractal dimension D_F of the set of inclusions, following a relation formerly predicted for saturated infiltration with sharp fronts. The moisture profiles show patterns of steps and plateaus that significantly change in time due to the self-similar distribution of segments of high and low conductivity, but almost equivalent profiles are obtained by rescaling with the same factor used in the fractal construction. If the choice of constitutive model parameters leads to low contrasts of unsaturated conductivities, the moisture profiles are smoothened but the time scaling of the infiltration length does not change. In the nearly saturated region, the equation of moisture evolution has a coefficient controlled by the saturated conductivity and a scaling approach shows how that conductivity varies with the position along the heterogeneous medium. This scaling implies that the same superdiffusive infiltration exponent 1/1+D_F is expected for regular and stochastic fractal distributions of low conductivity inclusions in conditions of unsaturated flow. The results suggest modeling some fire-clay bricks as layered media with low conductivity layers arranged as a fractal pattern with dimension of ≈0.63.