In this paper we propose the application of a new model of transients of pore
pressure p and solute density \r{ho} in geologic porous media. This model is
rooted in the non-linear waves theory, the focus of which is advection and
effect of large pressure jumps on strain (due to large p in a non-linear
version of the Hooke law). It strictly relates p and \r{ho} evolving under the
effect of a strong external stress. As a result, the presence of quick and
sharp transients in low permeability rocks is unveiled, i.e. the non-linear
Burgers solitons. We therefore propose that the actual transport process in
porous rocks for large signals is not the linear diffusion, but could be
governed by solitons. A test of an eventual presence of solitons in a rock is
here proposed, and then applied to Pierre Shale, Bearpaw Shale, Boom Clay and
Oznam-Mugu silt and clay. A quick analysis showing the presence of solitons for
nuclear waste disposal and salty water intrusions is also analyzed. Finally, in
a kind of "theoretical experiment" we show that solitons could also be present
in Jordan and St. Peter sandstones, thus suggesting the occurrence of osmosis
in these rocks
