The transition to renewable energy, marked by the intermittent nature of solar and wind sources, requires the production and storage of large quantities of hydrogen. The utilization of underground salt caverns is recognized as a proven technology for the storage of many materials, such as oil, natural gas, etc. From a geomechanical viewpoint, the main challenge for hydrogen storage in salt caverns is associated to the fast cyclic injection/production operations [1]. To design efficient and safe operations in such unusual conditions, numerical simulations are an important tool. However, mechanics of salt rocks is complex, and salt formations may be strongly heterogeneous, which makes challenging to building a reliable simulation framework. In this work, we develop a 3D finite element simulator based on FEniCS for the simulation of salt caverns. The constitutive model for salt mechanics considers a viscoelastic and a viscoplastic [2] model for primary creep, Norton's power law [3] model for steady-state creep, and damage mechanics for tertiary creep. The importance of each contribution in the constitutive model is discussed and analyzed. The work also presents a sensitivity analysis regarding heterogeneous mechanical properties on the cavern's performance. The results show that the viscoplastic contribution has a strong impact when the production of hydrogen exceeds a maximum limit. Additionally, the presence of heterogeneous layers may induce stress concentrations that may lead to unstable damage propagation (tertiary creep).