This Master thesis investigates the formation and evolution of subglacial drainage systems underneath ice sheets. The presence of water underneath ice masses has enormous effects on ice dynamics as it acts as a lubricant. Ice sheets play a big role in ocean circulation and climate dynamics. So far little is known about the formation and evolution of subglacial water. However, various elements of hydrological systems are suggested to exist ranging from thin water sheets to channels and lakes. This Master thesis attempts to fill this gap and creates a conceptual thermo-mechanical model of ice and water dynamics, which is solved using the finite element method. While the Stokes-equation is used for ice, the Navier-Stokes equation is introduced for the simulation of the water flow. Thus the acceleration terms are included. A coupling of water and ice flow has to overcome the problem that water flows on short time scales while ice on long once. This problem is solved with an adaptive timestepping mode. In order to evolve a water layer the melt rates are calculated by a state of the art enthalpy formulation, using the advantage of solving for temperature and water content at the once. The results demonstrate that it is possible to model both ice and water flow within one approach using the Navier-Stokes flow and the enthalpy formulation. The experiments investigate the effects of geothermal heat flux and inflow velocities on the melt rate behaviour. The results show that thin water layers develop into a channelized flow form when a certain threshold in the water velocity is overcome. Moreover, cavities form in regions of increased geothermal heat flux. With increasing inflow velocity the cavity propagates further in the direction of flow. The results demonstrate a strong relationship between water flow velocities and changes in melt dynamics. Further, it illustrates that the subglacial water system is under constant change. Hence, this study gives an important insight into melt dynamics under ice masses and their evolution.