Underground hydrogen storage facilities play a crucial role in facilitating the transition from fossil fuels to hydrogen energy. Accurately calculating the temperature and pressure distribution of hydrogen gas, which has low density and viscosity, during the injection and production processes is essential for preventing wellbore safety issues. This study develops a mathematical model to simulate the temperature and pressure variations within the wellbore during the injection and production processes, accounting for the Joule-Thomson effect of hydrogen gas. The results indicate that the significant temperature disparity within the wellbore during hydrogen injection may lead to deformation of the tubing and casing due to uneven thermal stress, with a relatively small impact from gas injection pressure on the temperature distribution. Within the flow rate range of 0.2 kg/s to 1.2 kg/s, an increase in flow rate results in a decrease in bottom-hole temperature, with the temperature varying between 356 K and 310 K. At various injection flow rates (0.4 kg/s, 0.8 kg/s, and 1.2 kg/s), hydrogen gas wells consistently exhibit lower temperatures than natural gas wells, whereas the reverse is true during the gas production process. These findings provide a foundation for adjusting hydrogen injection and production strategies to ensure the safe operation of gas storage facilities.