When a thin tube is dipped into water, the water will ascend to a certain height, against the action of gravity. While this effect, termed capillarity, is well known, recent experiments have shown that agitated granular matter reveals a similar behavior. Namely, when a vertical tube is inserted into a container filled with granular material and is then set into vertical vibration, the particles rise up along the tube. In the present Letter, we investigate the effect of granular capillarity by means of numerical simulations and show that the effect is caused by convection of the granular material in the container. Moreover, we identify two regimes of behavior for the capillary height H_{c}^{∞} depending on the tube-to-particle-diameter ratio, D/d. For large D/d, a scaling of H_{c}^{∞} with the inverse of the tube diameter, which is reminiscent of liquids, is observed. However, when D/d decreases down to values smaller than a few particle sizes, a uniquely granular behavior is observed where H_{c}^{∞} increases linearly with the tube diameter.