Abstract: We introduce an uncertainty-aware Digital Twin (DT) for monitoring, optimization, and decision making in support of underground storage operations, with a focus on Geological Carbon Storage (GCS). In real-world scenarios, forward models are often misspecified due to uncertainties in subsurface dynamics and observation models. Our DT addresses this challenge by incorporating context-awareness into its neural networks to account for complexities in indirect seismic observations, including variability in rock-physics relations that link the reservoir states (e.g., pressure and saturation) to time-lapse seismic responses. To achieve this, we employ sensitivity-aware amortized Bayesian inference (SA-ABI), a simulation-based inference method that integrates sensitivity analysis into the training phase. This enables the DT to quantify and propagate uncertainty stemming from model discrepancies, particularly in rock-physics parameters. Computational efficiency is maintained through shared neural network weights that