# Shape Modelling and Volume Optimisation of Salt Caverns for Energy Storage
## 2 Three-Dimensional Numerical Model and Assessment Criteria
### Method of Analysis and Assessment Criteria  
The evaluation of the cavern shapes and dimensions was based on numerical modelling. The 3D geomechanical model was introduced into FLAC\({}^{3\text{D}}\)--software for solving geomechanical and geotechnical problems, including rheological phenomena.  
The effect of the cavern shape and dimensions on cavern stability and volumetric closure was examined by the analysis of the following criteria: the displacements, effective strains (ESs), von Mises stress (vMS), strength/stress ratio (SSR) and safety factor (SF). The same criteria were used in the optimisation of the cavern shape and volume. Displacements were analysed at the cavern sidewalls as well as at the bottom and the roof of the caverns. The displacements also illustrated the decrease in cavern volume with time caused by the salt creep (volumetric closure). The vMS is the equivalent effective stress that is related to the creep rate within the primary and the secondary creep stages. ES Equation (1) is a scalar quantity that represents the magnitude of the plastic strain tensor. The ES is expressed by  
\[\varepsilon_{ef}=\sqrt{\frac{2}{3}\varepsilon^{dev}:\varepsilon^{dev}} \tag{1}\]  
where \(\varepsilon_{ef}\) is the effective strain and \(\varepsilon^{dev}\) is the deviatoric strain tensor. The effective strain represents damage associated with plastic deformation. The ES of the rock mass surrounding the cavern should be less than 3% during the period of operation, which is usually 30 years [29; 34]. The strength/stress ratio (SSR) is used to indicate the dangerous areas in the sidewalls of the caverns and in its surroundings. An SSR of 1.0 implies a material failure, and an SSR of 2 indicates that the material reaches 50% of its strength. The safety factor (SF) is a dilatancy damage criterion Equation (2) for rock salt that can be expressed by  
\[SF=\frac{b\cdot I_{1}}{\sqrt{I_{2}}} \tag{2}\]  
where \(b\) is a material constant, \(I_{1}\) is the first invariant of the stress tensor and \(I_{2}\) is the second invariant of the deviatoric stress tensor. According to [PERSON] et al. [38], [PERSON] et al. [39; 40] and [PERSON] [26], an SF < 1 indicates failure, and an SF < 1.5 shows local damage.  
In addition, the volumetric closure (volume convergence) of each cavern shape was evaluated. The volumetric closure was calculated as the ratio of the volume loss to the initial cavern volume. The accepted convergence rate was below 1% per year [41].  
The numerical analysis was performed for a period of 9.5 years and consisted of three phases: leaching, 4 years when the caverns were filled with brine, and five operation cycles in which the cavern pressure oscillated between the minimum of 4 MPa to the maximum of 17.5 MPa. For each operation cycle, the aforementioned criteria were analysed in two periods: at the end of the gas injection period, and at the end of the gas withdrawal period. The cavern leaching was simulated to be excavated instantly in the applied software. This assumption resulted from the fact that the displacements in the leaching period were very small in comparison to the operation period. Consequently, their role in the long-term cavern stability was neglected. To eliminate the adverse effect of this assumption on the results, the in situ stress redistribution was calculated first and used as the initial conditions for further calculations including creep.