Reply to the Comment by Christopher Hall

We thank C. Hall for the Comment on imbibition in other material contexts. Two different communities can consider some aspects of the same problem, but with different approaches and hopefully new insights. We recognize that we missed the rich literature from the soil physics and construction materials communities and thank C. Hall for his valuable description of past research. However, our letter concerns model systems, which lead to specific dynamics not documented as well in these other literatures. Specifically, in our paper, we consider two ideal porous systems that we describe with analytical models derived from classical assumptions. We consider only perfect wetting conditions realized by using silicon oil as the liquid and glass spheres as the material. These glass spheres constitute an ideal porous material by having wellcalibrated sizes and shapes. In the first part of our letter, we are interested in imbibition in two uniform porous layers and we identify conditions and times during which the wetting front in the second layer advances with a constant velocity, which is clearly an unusual dynamical response in imbibition phenomena. This condition is realized by using two-layer systems made of small beads (of diameter ds) over a given length s and then large beads (of diameter d ). We show that as the liquid penetrates into the second layer, there is a well-characterized distance in which the imbibition front advances linearly in time. This regime is due to localized viscous dissipation into the layer of small beads, which can dominate the dissipation in the layer of large beads. The condition ds d is essential to observe this regime over a sufficiently long time and cannot be observed otherwise. We note that this regime can be discerned in fig. 7 of [1], but is not described as a linear regime by the authors. One of the aims of our paper consisted of pointing out this particular regime, which is completely different from what is commonly accepted and is not well documented in the soil physics or other literatures, to the best of our knowledge. An interesting feature of this unusual linear regime is its possibility to be perfectly predicted by choosing the ratio of the bead diameters and the length of the initial layer made of the small beads (eq. (12) of our paper [2]). We think that this linear response can be useful in other contexts such as analytical sciences, for example. In the second part of our paper, we analyze another particular case of layered materials where the layers are not chosen arbitrarily but organized in a way that the gradient of permeability is approximately constant over the sample. The experimental results are compared to the analytical model, with both in dimensionless form, and very good agreement is observed. The case treated in our letter differs from the ones from refs. [3,4], in that we are able to find an analytical solution that is an excellent description of the results of our model experiments. To conclude, we recognize that we have missed literature from the soil physics and construction materials communities, but we think that our work integrating layered responses and controlled gradients provides added value, and in particular points out a linear response regime that had not been documented before in the literature.