  We study the forced fluid invasion of an air-filled model porous medium at
constant flow rate, in 1+1 dimensions, both experimentally and theoretically.
We focus on the non-local character of the interface dynamics, due to liquid
conservation, and its effect on the scaling properties of the interface upon
roughening. Specifically, we study the limit of large flow rates and weak
capillary forces. Our theory predicts a roughening behaviour characterized at
short times by a growth exponent $\beta_1 = 5/6$, a roughness exponent
$\alpha_1=5/2$, and a dynamic exponent $z_1=3$, and by $\beta_2=1/2$,
$\alpha_2=1/2$, and $z_2=1$ at long times, before saturation. This theoretical
prediction is in good agreement with the experiments at long times.The ensemble
of experiments, theory, and simulations provides evidence for a new
universality class of interface roughening in 1+1 dimensions.
