We employ nuclear magnetic resonance imaging to study water penetration in cylindrical blocks of unsized paper prepared under different molding pressures. From the measured kinetics of the imbibition profiles, we determine the dependence of the effective transport diffusivity upon degree of saturation of the pores by the penetrating fluid. In general, the transport process is found to be non-Fickian and we discuss different methods of data analysis adapted to this situation. The effective transport diffusivity vividly captures the presence of a precursor front, consisting of fluid in partially filled pores, with a much higher effective diffusivity than that of fluid in largely saturated pores.