A model of imbibition dynamics in a channel of flattened triangular cross-section is presented, taking into account the liquid film flow in the corners of the channel. The quasi-analytical solutions are derived on the basis of a lubrication approximation. The analysis encompasses two imbibition scenarios corresponding to a constant flow rate or constant pressure imposed in the wetting fluid at the inlet of the channel. In the former case, the process starts with a liquid film flow regime in the corners that is followed by a bulk and corner film flow regime characterised by a triple point advancing (far) ahead of the bulk meniscus after its entrance in the channel. In the latter case, the occurrence of the bulk and corner film flow regime is conditioned by an imposed pressure yielding a capillary pressure at the inlet smaller than a threshold capillary pressure. Above this threshold, the liquid film regime remains. For both imbibition scenarios under concern, important features are highlighted, including (i) the time scalings of the dynamics of both the triple point and apex of the bulk meniscus (when it exists), (ii) the contrast in the positions of these two points showing that the classical Washburn approach, which neglects the effect of the corner films, overpredicts the dynamics of the bulk meniscus. The important consequence is an early wetting fluid breakthrough at the channel outlet much before the bulk meniscus arrival. Comparisons with experimental data available in the literature are provided, validating the approach proposed in this work.