Natural gas hydrate (NGH) is a kind of new type green energy source with giant reserves which has been thought of highly by energy explorers in the world. However, NGH breaks down to produce some natural gas that enters the annulus and flows together with the drilling fluid. The gas-liquid two-phase flow can have an impact on the work of the drill string. Therefore, it is important to study gas-liquid two-phase flow in the annulus on the dynamic characteristics of the drill string. In this article, taking a single drill string as the research object, a fluid-structure coupled finite element mathematical model of two-phase flow in the annulus and drill string is established based on computational fluid dynamics and computational structural dynamics theory. The finite element numerical simulation method is used to analyze the influence of drilling fluid and natural gas in the annulus on the dynamic characteristics of the drill string. The simulation analysis shows the following: (1) The motion of drilling fluid or natural gas in the annulus will reduce the natural frequency of the drill string, and the drilling fluid has a greater impact on the natural frequency of the drill string. (2) When single-phase drilling fluid flows in the annulus, the displacement peak in different directions, maximum equivalent stress, and strain of the drill string increase with the increase of the drilling fluid flow velocity or pressure, and the drilling fluid pressure has a more significant effect. (3) When the gas-liquid two-phase fluid flows in the annulus, the displacement peak, maximum equivalent stress, velocity amplitude, and acceleration amplitude of the drill string all increase with the natural gas flow velocity and natural gas content increase, and the natural gas flow velocity has a more significant effect.