Reservoir simulation is crucial in understanding and predicting the behavior of subsurface reservoirs, aiding in efficient resource management. By modeling fluid flow, pressure changes, and other dynamic processes, it provides valuable insights for optimizing development strategies. This helps in making informed decisions, minimizing risks, and maximizing the sustainable use of natural resources such as water, geothermal energy, or carbon storage. However, simulating fluid flow in porous media is computationally intensive; accurately modeling a single injection scenario for a large CO 2 reservoir can take several hours on a high-performance computing cluster. This becomes a bottleneck when running numerous simulations, such as in the 'Uncertainty Quantification' process. For any context involving reservoir simulation (such as CO 2 storage, Hydrogen storage, or oil extraction), well events, including their opening and closure, introduce significant numerical challenges due to their immediate effects on pressure and saturation variables. This often necessitates a drastic reduction in time step size to solve the non-linear equations derived from the discretization of the continuous mathematical model. Despite this, the impact of these specific well events tends to be relatively similar over space and time. We propose a local preconditioning strategy in the near-well region to mitigate the impact of well events. We complement the standard fully implicit solver by predicting an initialization in the nearwell region of Newton's method using supervised learning. More specifically, we replace the initialization in pressure by a linear approximation obtained