We perform boundary integral simulations to understand the removal of Newtonian liquids from a model gravure cell. Two different configurations are considered. In the first configuration, there is a free surface and an outlet boundary, and the liquid is driven out of a cavity by a combination of horizontal substrate motion and an imposed pressure gradient; a similar model was used by Powell [Trans. IChemeE, Part C 78, 61 (2000)]. The percentage of liquid remaining in the cavity Vr is influenced by the capillary number Ca, cavity depth D, and contact angle θ. We found that Vr decreases with a decrease in Ca or D, consistent with prior studies, and for a shallow enough cavity, almost all of the liquid can be removed. Additionally, Vr decreases with an increase in θ. In the second configuration, there are two free surfaces, and the liquid is driven out of the cavity by moving the substrate both horizontally and vertically. Our simulations suggest that Vr decreases with an increase in the extensional velocity V, and in some cases the entire cavity can be emptied when V is greater than a critical value. The present work sheds light on the roles that surface wettability, cavity size, substrate kinematics, and free-surface dynamics play in surface-tension-driven liquid emptying from tiny cavities.