Among the tremendous amount of work that exists on spontaneous imbibition, the broadening of a fluid-fluid interface in a set of parallel fibers has not been tackled yet. Nevertheless, this problem is relevant to various applications. In a seminal paper, Princen derived a model which predicts the conditional existence of capillary bridges between pairs of fibers in remarkable lattices at the hydrostatic equilibrium. In the present work, it is argued that this model provides reliable predictions in dynamics. Therefore, the growth of these capillary bridges in a regular square lattice of parallel fibers has been studied through direct numerical simulations. In a dimensionless framework, the influence of the spacing of fibers, inertia, and viscosity and density ratios were investigated. The results show that Princen’s predictions are in quantitative agreement with the observations. Furthermore, the temporal growths of capillary bridges are in qualitatively agreement with the Bosanquet’s model. From the sensitivity study, it appears that the spacing parameters defines the pressure difference which drives the growth of capillary bridges. Inertia plays an insignificant role in the dynamics compared to the imbibition of a circular capillary. The non-wetting phase has a negligible influence on the flow in a liquid-gas system. Finally, a simulation in a complex disordered set of parallel fibers has been done. In such a system, the predictions of Princen’s and Bosanquet’s models are still in qualitative agreement with the numerical simulations.