  Recent experiments of imbibition in columnar geometries show interfacial
fluctuations whose dynamic scaling is not compatible with the usual non local
model governed by surface tension that results from a macroscopic description.
To explore this discrepancy, we exhaustively analyze numerical integrations of
a phase-field model with dichotomic columnar disorder. We find that two
distinct behaviors are possible depending on the capillary contrast between
both values of disorder. In a high contrast case, where interface evolution is
mainly dominated by the disorder, an inherent anomalous scaling is always
observed. Moreover, in agreement with experimental work, the interface motion
has to be described through a local model. On the other hand, in a lower
contrast case, interface is dominated by interfacial tension and can be well
modeled by a non local model. We have studied both spontaneous and forced-flow
imbibition situations, giving a complete set of scaling exponent in each case,
as well as, a comparison to the experimental results.
