Hydrate formation in foam drainage gas recovery wells and the shut in accidents caused by plugging have become an important problem that restricts the safe production of natural gas. The blockage and accumulation of hydrates is a gradual problem. This research goes beyond predicting the formation of hydrates and delves deeper into examining the rate of hydrate formation and the degree of pipeline blockage at different wellbore locations. First, the temperature model, pressure model, multiphase flow model, and hydrate plugging model of hydrate formation process are established from the equations of mass conservation, energy conservation, and momentum conservation. Second, an iterative approach is employed to solve the model, with a maximum error of 6.86% in model validation. Finally, sensitivity analysis shows that wellhead temperature, wellhead pressure, and foam viscosity have different effects on hydrate formation, maximum plugging position, and plugging degree. At the same time, combined with the actual drainage and gas production process, and the characteristics of hydrate blockage, proposed hydrate prevention measures can be taken to achieve safe production of natural gas. The research results indicate that a decrease in temperature signifies an increase in undercooling, resulting in an accelerated rate of hydrate formation and an elevated risk of hydrate blockage. The decrease in wellhead pressure leads to a decrease in the rate of hydrate formation and an increase in production, which is beneficial for the hydrate prevention. However, larger pressure differences and gas production rates will put higher requirements on equipment such as well control devices. An increase in foam viscosity will lead to increased pressure, foam compression, reduced drainage capacity, and intensified hydrate generation. Therefore, foam viscosity should be kept as small as possible to keep the foam stable.