The present disclosure relates to methods and apparatuses for removing water from a wet fibrous web. During the process of making a fibrous structure, a capillary dewatering apparatus remove water from a wet porous web. In some configurations, a capillary dewatering apparatus may include a capillary porous media. A molding member, such as a papermaking belt comprising an air permeable fabric, may advance the wet fibrous web onto the capillary porous media, wherein the fibrous web is positioned between the capillary porous media and the air-permeable fabric. An energy transfer surface may be positioned in contact with the air-permeable fabric or the outer circumferential surface, wherein the energy transfer surface operates to vibrate the capillary porous media. In turn, the vibration helps to drive water through the capillary porous media, allowing additional water to flow from the fibrous web and through pores in the capillary porous media.