  We propose a theory for erosional channelization induced by fluid flow in a
saturated granular porous medium. When the local fluid flow-induced stress is
larger than a critical threshold, grains are dislodged and carried away so that
the porosity of the medium is altered by erosion. This in turn affects the
local hydraulic conductivity and pressure in the medium and results in the
growth and development of channels that preferentially conduct the flow. Our
multiphase model involves a dynamical porosity field that evolves along with
the volume fraction of the mobile and immobile grains in response to fluid flow
that couples the spatiotemporal dynamics of the three phases. Numerical
solutions of the resulting initial boundary value problem show how channels
form in porous media and highlights how heterogeneity in the erosion threshold
dictates the form of the patterns and thus the ability to control them.
