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Drilling in a natural gas hydrate formation is challenging due to the poor consolidation of the formation and the potential evaporation of the hydrate. The unreasonable down-hole pressure of the drilling fluid can not only lead to the wellbore instability, but also change the predrilling condition of the natural gas hydrate formation, thus leading to an instable wellbore. In this paper, the integrated discrete element method (DEM)-computational fluid dynamics (CFD) work flow is developed to study the wellbore instability due to the penetration of the drilling fluid into the hydrate formation during crack propagations. The results show that the difference between in-situ stresses and overpressure directly affect the drilling fluid invasion behavior. The lower hydrate saturation leads to an easier generation of drilling fluid flow channels and the lower formation breakdown pressure. The breakdown pressure increases with the increase of hydrate saturation, this also indicates that hydrates can enhance the mechanical properties of the formation. The induced cracks are initially accompanied with higher pressure of the drilling fluid. According to the rose diagram of the fracture orientation, a wider orientation of the fracture distribution is observed at higher pressure of the invasion fluid.