A comprehensive analytical and experimental framework is presented to
describe gravity-driven motions of rheologically complex fluids through
porous media. These phenomena are relevant in geophysical,
environmental, industrial and biological applications.
The fluid is characterized by an Ostwald-DeWaele constitutive equation with
behaviour index n. The flow is driven by the release of fluid at the origin of
an infinite porous domain. In order to represent several possible spreading
scenarios, we consider: i) different domain geometries: plane, radial, and
channelized, with the channel shape parameterized by ; ii) instantaneous
or continuous injection, depending on the time exponent of the volume of
fluid in the current, ; iii) horizontal or inclined impermeable boundaries.
Systematic heterogeneity along the streamwise and/or transverse direction
is added to the conceptualization upon considering a power-law
permeability variation governed by two additional parameters and .
Scalings for current length and thickness are derived in self similar form
coupling the modified Darcy’s law accounting for the fluid rheology with the
mass balance equation. The speed, thickness, and aspect ratio of the
current are studied as a function of model parameters; several different
critical values of emerge and govern the type of dependency, as well as
the tendency of the current to accelerate or decelerate and become thicker
or thinner at a given point. The asymptotic validity of the solutions is limited
to certain ranges of model parameters.
Experimental validation is performed under constant volume, constant and
variable flux regimes in tanks/channels filled with transparent glass beads
of uniform or variable diameter, using shear-thinning suspensions and
Newtonian mixtures. The experimental results for the length and profile of
the current agree well with the self-similar solutions at intermediate and late
times
