High temperature infiltration at low overpressures: Darcy's law, the slug-flow hypothesis and percolation

Experiments on liquid metal infiltration into porous preforms at low overpressures give a linear relationship between the square of the infiltrated height and the applied over-pressure. This result can be derived from Darcy's law under the Slug Flow Hypothesis SFH. Two features characterize SFH: (i) a step-like drainage curve, i.e., homogeneous, not necessarily full, filling of the empty space, and (ii) a linear drop of pressure through the infiltrated sample. However, experimental data do also indicate that, in most cases, (i) is not fulfilled. In this work, going beyond SFH, we utilize several combinations of drainage curve (Brooks and Corey, Van Genuchten and percolation) and permeability (Mualem, Burdine and a power law) to investigate whether the linear relationship may show up even though the SFH is not fulfilled. We show that, at low over-pressures, the integro-differential equation which describes this system admits a power law solution whose exponent and constant can be analytically related to the model parameters. This allows to predict that all combinations, except those including Burdine permeability, reproduce that linear relationship. In addition, the remaining six give a proportionality coefficient 1 as in SFH, actually is equal to 1 only for full filling (in the case of Mualem the coefficient of the drainage curve has to be 1). However, only the two combinations based upon Percolation have a drainage curve with an exponent that can be less than 1, in agreement with recent experimental studies. Finally, albeit the drainage curve is not a step function, pressure approximately varies linearly throughout the infiltrated sample. The present analysis and methodology may be of help in a variety of fields such as soil science, oil extraction, hydrology, geophysics, metallurgy, etc.