Inferring filtration laws from the spreading of a liquid modelling by the porous medium equation

Motivated by modelling the spreading of a two-dimensional particle-laden gravity current on a porous membrane, we couple the porous medium equation with drainage with a blocking law for the pores of the membrane. The blocking law characterises a range of blocking phenomena through the choice of a single blocking parameter α ∈ [−∞, ∞]. We pose the question whether the value of the blocking parameter, and hence the blocking law, can be inferred by observing the position of the front of the current over time. We use two different strategies to determine the blocking parameter for almost the entire possible range of values. First, we show that the position of the front follows a power law when the current is fed by a constant influx at the center and that the exponent of the power law is unique for sufficiently large blocking parameters. Second, we show that the coupled system of the porous medium equation with absorption and the blocking law allows for a travelling wave solution if a suitable influx, dependent on the blocking parameter, is applied. For α < 1, we show that the suitable influx is, to leading order, independent of α and that the corresponding travelling wave has a finite region in which fluid drains at the front and whose width can be used to infer the blocking parameter.