Constitutive functions that govern macroscale capillary pressure and relative permeability are central in constraining both storage efficiency and sealing properties of CO$_2$ storage systems. Constitutive functions for porous systems are in part determined by wettability, which is a pore-scale phenomenon that influences macroscale displacement. While wettability of saline aquifers and caprocks are assumed to remain water-wet when CO$_2$ is injected, there is recent evidence of contact angle change due to long-term CO$_2$ exposure. Weakening of capillary forces alters the saturation functions dynamically over time. Recently, new dynamic models were developed for saturation functions that capture the impact of wettability alteration (WA) due to long-term CO$_2$ exposure. In this paper, these functions are implemented into a two-phase two-component simulator to study long-term WA dynamics for field-scale CO$_2$ storage. We simulate WA effects on horizontal migration patterns under injection and buoyancy-driven migration in the caprock. We characterize the behavior of each scenario for different flow regimes. Our results show the impact on storage efficiency can be described by the capillary number, while vertical leakage can be scaled by caprock sealing parameters. Scaling models for CO$_2$ migration into the caprock show that long-term WA poses little risk to CO$_2$ containment over relevant timescales.