  We show that when a non-wetting fluid drains a stratified porous medium at
sufficiently small capillary numbers Ca, it flows only through the coarsest
stratum of the medium; by contrast, above a threshold Ca, the non-wetting fluid
is also forced laterally, into part of the adjacent, finer strata. The spatial
extent of this partial invasion increases with Ca. We quantitatively understand
this behavior by balancing the stratum-scale viscous pressure driving the flow
with the capillary pressure required to invade individual pores. Because
geological formations are frequently stratified, we anticipate that our results
will be relevant to a number of important applications, including understanding
oil migration, preventing groundwater contamination, and sub-surface CO$_{2}$
storage.
