  We study the impact of the wetting properties on the immiscible displacement
of a viscous fluid in disordered porous media. We present a novel pore-scale
model that captures wettability and dynamic effects, including the
spatiotemporal nonlocality associated with interface readjustments. Our
simulations show that increasing the wettability of the invading fluid (the
contact angle) promotes cooperative pore filling that stabilizes the invasion,
and that this effect is suppressed as the flow rate increases, due to viscous
instabilities. We use scaling analysis to derive two dimensionless numbers that
predict the mode of displacement. By elucidating the underlying mechanisms, we
explain classical yet intriguing experimental observations. These insights
could be used to improve technologies such as hydraulic fracturing, CO$_{2}$
geo-sequestration, and microfluidics.
