Nearly all meltwater from glaciers and ice sheets is routed englacially through moulins. Therefore, the geometry and evolution of moulins has the potential to influence subglacial water pressure variations, ice motion, and the runoff hydrograph delivered to the ocean. We develop the Moulin Shape (MouSh) model, a time-evolving model of moulin geometry. MouSh models ice deformation around a moulin using both viscous and elastic rheologies and melting within the moulin through heat dissipation from turbulent water flow, both above and below the water line. We force MouSh with idealized and realistic surface melt inputs. Our results show that variations in surface melt change the geometry of a moulin by approximately 20% daily and by over 100% seasonally. These size variations cause observable differences in moulin water storage capacity, moulin water levels, and subglacial channel size compared to a static, cylindrical moulin. Our results suggest that moulins are important storage reservoirs for meltwater, with storage capacity and water levels varying over multiple timescales. Implementing realistic moulin geometry within subglacial hydrologic models would therefore improve the representation of subglacial pressures, especially over seasonal periods or in regions where overburden pressures are high.