Filaments are common structures in the solar atmosphere, and usually interact with their surrounding magnetic fields. However, interactions between filaments and network fields are rare. Here, we report interactions between filament fibrils and a nearby network field in the quiet Sun by employing observations from the New Vacuum Solar Telescope (NVST) and Solar Dynamics Observatory. NVST Hα images show that several filament fibrils separated from the main body of the filament, and moved sideward. While a fibril met the network field, the movement of the fibril segment corresponding to the network field slowed down. Subsequently, weak extremely ultraviolet brightenings appeared near the interface of the filament and the network field, and then the fibril materials began to converge toward the network field. Meanwhile, continuous redshift signal enhancements appeared in the corresponding Dopplergrams, accompanying the convergences of the fibril materials. About 10 and 35 minutes later, two other similar processes occurred again. These observations imply that the network field blocks movements of the filament fibrils and weak magnetic reconnections between the blocked fibrils and the network field take place. We suggest that new field lines developed due to the magnetic reconnections, along which fibril materials fell down into the lower solar atmosphere. These results provide a new picture of filament material drainage.