Unmanned aerial-underwater vehicles (UAUVs) provide the potential for working on missions in complex multi-domain environments. To achieve amphibian mobility, current UAUV designs rely on additional mechanical components such as multiple layers of propeller blades, water ballast, buoys or wings. This paper presents a miniature UAUV which has a simple mechanical design that resembles a traditional quadcopter. The paper discusses the dynamic modelling, state estimation and control strategy for this UAUV, as well as a detailed characterization of the quadcopter blades operating in the air and water regimes. A strategy for the UAUV to breach calm water surface is then proposed and experimentally tested. The results demonstrate that the UAUV can successfully breach the still water surface, but also show tracking error and breaching delay that are not fully characterized by the model. This suggests the need to carry out further analysis on the dynamics of the UAUV both underwater and in the transition regime.