Supercritical CO2 injected into a saline aquifer will eventually dissolve in the formation waters. The density of formation waters increases with CO2 concentration and becomes negatively buoyant. In time convective instabilities could form and move CO2 downwards away from the caprock. This is favourable for carbon capture and storage as it reduces the risk of CO2 escaping through the caprock. The presence of a leaky fracture in the caprock could significantly jeopardise the security of a CO2 storage complex. However, lateral diffusion and subsequent convection of CO2 out of the fracture could increase overall dissolution rates. This paper presents a set of numerical simulations, using the TOUGH2-MP/ECO2N reservoir simulator, that explore the potential for convection enhanced CO2 dissolution from leaking fractures. Simulations have been performed for varying fracture inclinations and Rayleigh numbers. We find that dissolution per unit area of fracture is increased for vertical fractures although total dissolution is increased for a larger fracture surface area. Also, higher Rayleigh number systems have higher mass transfer rates.