  During the past sixty minutes only, oil companies have extracted six
trillions liters of oil from the ground, i.e. the volume of about two hundreds
Olympic swimming pools. This phenomenal number gives a striking illustration of
the impact of multiphase flows on the world economy and environment. From a
fundamental perspective, we now clearly understand the large-scale patterns
formed when liquid interfaces are driven through heterogeneous environments. In
stark contrast, the displacement of fragmented fluids through disordered media
remains limited to isolated droplets and bubbles. Here, we elucidate the
collective dynamics of emulsions hydrodynamically driven through disordered
environments. Advecting hundreds of thousands of microfluidic droplets through
random lattices of pinning sites, we establish that the mobilization of
confined emulsions is a critical dynamical transition. Unlike contact-line
depinning, emulsion mobilization is not triggered by large-scale avalanches but
merely requires the coordinated motion of small groups of particles.
Criticality arises from the correlations of seemingly erratic depinning events
over system-spanning scales along smectic river networks. We elucidate the
microscopic origin of these self-organized flow patterns: contact interactions
and hydrodynamic focusing conspire to mobilize emulsion out of disorder. We
close our article commenting on the similarities (and profound differences)
with the plastic depinning transitions of driven flux lines in high-$T_{\rm c}$
superconductors, and grain transport in eroded sand beds.
