A novel 2D finite element method (FEM) on unstructured grid for nonlinear time-dependent deformation of materials is developed. The objective is to model the complex deformation behavior of the rock salt, inside which caverns are mined to store green fuels (such as hydrogen). The analyses and the developments of the present work allow for quantification of the state of the stress of the cavern, and assess the safety and reliability of the storage structure over time. The novelty of the approach is in using minimization of the potential energy principle in conjunction with nonlinear creep deformation physics. While the available FEM-based simulators offer tools only for solving linear and non-linear elastic models, the developed simulator takes into account cyclic loading and material’s damage evolution in time with possibility to predict the material’s failure. Apart from that, the stored product (hydrogen) density is taken into account, which affects cavern’s pressure variation with depth. Impurities,causing heterogeneous rock salt properties, are also considered in the developed model. Multivariate Gaussian distribution is utilized to generate distribution of the heterogeneous mechanical properties. Eulerian strains are introduced in the model to take into account deformation of the mesh. As such,the computational grid changes its geometry according to the deformation. Several numerical test cases are studied. Firstly, a consistency (verification) study is performed, to validate the linear elastic model. Remark that the linear elastic deformation model casts the basis of the non-linear model with creep physics. Then, several studies have been performed to analyze, quantify and approximate the deformation of the salt cavern under the gas pressure change, including the time-dependent creep physics.