The Greenland Ice Sheet (GrIS) is losing mass as the climate warms through both increased meltwater runoff and ice discharge at marine-terminating sectors. At the ice sheet surface, meltwater runoff forms a dynamic supraglacial hydrological system which includes stream and river networks and large supraglacial lakes (SGLs). Streams and rivers can route water into crevasses or into supraglacial lakes with crevasses underneath, both of which can then hydrofracture to the ice sheet base, providing a mechanism for the surface meltwater to access the bed. Understanding where, when, and how much meltwater is transferred to the bed is important because variability in meltwater supply to the bed can increase ice flow speeds, potentially impacting the hypsometry of the ice sheet in grounded sectors, and iceberg discharge to the ocean. Here we present a new, physically based, supraglacial hydrology model for the GrIS that is able to simulate (a) surface meltwater routing and SGL filling; (b) rapid meltwater drainage to the ice sheet bed via the hydrofracture of surface crevasses both in and outside of SGLs; (c) slow SGL drainage via overflow in supraglacial meltwater channels; and, by offline coupling with a second model, (d) the freezing and unfreezing of SGLs from autumn to spring. We call the model the Supraglacial Hydrology Evolution and Drainage (or SHED) model. We apply the model to three study regions in southwest Greenland between 2015 and 2019 (inclusive) and evaluate its performance with respect to observed supraglacial lake extents and proglacial discharge measurements. We show that the model reproduces 80 % of observed lake locations and provides good agreement with observations in terms of the temporal evolution of lake extent. Modelled moulin density values are in keeping with those previously published, and seasonal and inter-annual variability in proglacial discharge agrees well with that which is observed, though the observations lag the model by a few days since they include transit time through the subglacial system, while the model does not. Our simulations suggest that lake drainage behaviours may be more complex than traditional models suggest, with lakes in our model draining through a combination of both overflow and hydrofracture and with some lakes draining only partially and then refreezing. This suggests that, in order to simulate the evolution of Greenland's surface hydrological system with fidelity, a model that includes all of these processes needs to be used. In future work, we will couple our model to a subglacial model and an ice flow model and thus use our estimates of where, when, and how much meltwater gets to the bed to understand the consequences for ice flow.