  We consider the dynamics and kinetic roughening of wetting fronts in the case
of forced wetting driven by a constant mass flux into a 2D disordered medium.
We employ a coarse-grained phase field model with local conservation of
density, which has been developed earlier for spontaneous imbibition driven by
a capillary forces. The forced flow creates interfaces that propagate at a
constant average velocity. We first derive a linearized equation of motion for
the interface fluctuations using projection methods. From this we extract a
time-independent crossover length $\xi_\times$, which separates two regimes of
dissipative behavior and governs the kinetic roughening of the interfaces by
giving an upper cutoff for the extent of the fluctuations. By numerically
integrating the phase field model, we find that the interfaces are superrough
with a roughness exponent of $\chi = 1.35 \pm 0.05$, a growth exponent of
$\beta = 0.50 \pm 0.02$, and $\xi_\times \sim v^{-1/2}$ as a function of the
velocity. These results are in good agreement with recent experiments on
Hele-Shaw cells. We also make a direct numerical comparison between the
solutions of the full phase field model and the corresponding linearized
interface equation. Good agreement is found in spatial correlations, while the
temporal correlations in the two models are somewhat different.
