<p>In this study, we address the challenge of stabilizing a numerical model for a coupled wellbore–reservoir multiphase hydrodynamic system, with a focus on optimizing wellbore cleanup operations in the oil and gas industry. This goal requires executing multiple forward simulations within optimization routines, elevating the importance of model stability and computational efficiency. The model used in this work is unique in its ability to directly simulate the cleanup of a near-wellbore zone of the hydrocarbon-filled porous reservoir formation damaged by the drilling mud invasion during drilling. Rather than relying on the conventional practice of reducing the relaxation factor to ensure a contractive coupling operator, we propose techniques that target intrinsic features of the solution to enhance stability. First, we implement adaptive time-stepping to improve robustness in temporal evolution. Second, we introduce a novel reservoir splitting approach for one-dimensional reservoir models, where flow rate is computed as the product of the flow rate per unit length and thickness. We transition to a multilayer reservoir structure, aligning each wellbore numerical cell with a distinct reservoir layer. A detailed description of this method is provided, along with simulation results demonstrating its impact on fidelity and stability. Third, we apply a phase correction mechanism to eliminate unphysical negative volume fractions arising before convergence. Finally, we enhance convergence using the Aitken relaxation method. These techniques collectively reduce the Lipschitz constant of the coupling operator, enabling both improved stability and faster convergence. We show that this reduction also offsets the computational overhead introduced, achieving an effective balance between robustness and efficiency.</p>