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Subsurface CO2 sequestration plays a crucial role in advancing carbon neutrality and supporting the transition to sustainable energy. However, the unique behavior of CO2, particularly during cold CO2 injection into depleted hydrocarbon reservoirs, poses challenges to wellbore injectivity. Addressing these challenges requires a numerical model that captures the complex interplay between wellbore dynamics and reservoir processes. In this work, we present DARTS-well, an open-source, fully coupled wellbore-reservoir model developed using the Operator-Based Linearization (OBL) technique. To this end, a transient, multi-segment, two-phase, non-isothermal wellbore model based on the Drift-Flux Model (DFM) is first developed and then coupled with the Delft Advanced Research Terra Simulator (DARTS) as the reservoir simulator. The model is demonstrated through test cases involving CO2 injection into a depleted reservoir, illustrating its potential to enhance the design and optimization of subsurface CO2 disposal systems.