  The migration of liquids in porous media, such as sand, has been commonly
considered at high saturation levels with liquid pathways at pore dimensions.
In this letter we reveal a low saturation regime observed in our experiments
with droplets of extremely low volatility liquids deposited on sand. In this
regime the liquid is mostly found within the grain surface roughness and in the
capillary bridges formed at the contacts between the grains. The bridges act as
variable-volume reservoirs and the flow is driven by the capillary pressure
arising at the wetting front according to the roughness length scales. We
propose that this migration (spreading) is the result of interplay between the
bridge volume adjustment to this pressure distribution and viscous losses of a
creeping flow within the roughness. The net macroscopic result is a special
case of non-linear diffusion described by a superfast diffusion equation (SFDE)
for saturation with distinctive mathematical character. We obtain solutions to
a moving boundary problem defined by SFDE that robustly convey a time power law
of spreading as seen in our experiments.
