The imbibition of water into cellulose fiber-based materials is studied with focus on the regime dominated by contact line dynamics rather than hydrodynamic drag of the bulk porous structure. Capillary rise is studied under the influence of gravity for different paper grades with a wide range of porosities. It is found that the capillary rise is logarithmic in time in the limit of long time-scales. This behavior is in excellent agreement with the Molecular Kinetics Theory (MKT) for contact line dynamics. For high-porosity paper grades, this previously neglected logarithmic regime starts already at about $5$\,cm of capillary intrusion, underscoring the critical importance of the contact line dynamics to the performance of cellulosic absorbents.