The structure of wellbore and coupling is connected by thread, it is the position where fatigue failure accidents are happened in gas storage well, and the make-up torque of thread is the key factor that influences the mechanical properties of this structure. In this pa- per, a finite element method was established to discuss the calculation method of make-up torque and the influence law on the stress of coupling in the gas storage well. Results show that the make-up torque cannot be simply ignored due to its great impact on internal stress distribution of gas storage well. Experimental results showed the correctness of the model. Under the action of the make-up torque, the overall stress level inside the wellbore is higher than the external coupling. However the maximum stress exists in the external coupling. The overall stress grad- ually increases with the increase of interference rotation number (abbr. RN) between coupling and wellbore, but the overall distribution trend is almost the same. The first thread on the left of the connection part is the area where fatigue failure is most likely to occur. The make-up torque between threads increases linearly with the increase of RN between cou- pling and wellbore.