Subsurface CO₂ sequestration is a promising method to advance carbon neutrality and support the shift toward sustainable energy. However, the unique behavior of CO₂ in these operations, particularly for cold CO₂ injection in depleted hydrocarbon reservoirs, poses challenges to wellbore injectivity, reservoir containment, and reservoir capacity. These challenges necessitate the development of a numerical model to better understand and optimize the interplay between wellbore dynamics and reservoir processes. In this work, we present the development of an open-source coupled wellbore-reservoir numerical model, named DARTS-well, which is tailored to CO₂ disposal in subsurface reservoirs. To this end, a multi-segment, multi-phase, non-isothermal wellbore model is first developed using the Drift-Flux Model (DFM), and its results for selected CO2 injection scenarios are validated against the commercial transient wellbore simulator OLGA. The multi-segment wellbore model is then coupled with the Delft Advanced Research Terra Simulator (DARTS) which is used in this study as the reservoir simulator. DARTS is widely used and validated for energy transition applications. The coupled model utilizes the Operator-Based Linearization (OBL) technique, employing state-dependent operators for thermodynamic properties interpolated from predefined tables or generated on the fly. This OBL parametrization approach addresses challenges associated with complex physics and reduces computational time, making it well-suited for modeling subsurface CO₂ sequestration.