# Optimization of Offshore Saline Aquifer CO2 Storage in Smeaheia Using Surrogate Reservoir Models
## 3 Methodology
### CO\({}_{2}\) Storage Optimization via Proxy Modeling  
According to the CO\({}_{2}\) storage goal, the objective of the Smeaheia project can be defined as follows:  
* To maximize the amount of CO\({}_{2}\) injected while maintaining a safe pressure level at the top of the injection area to avoid fractures in the caprock and to prevent the CO\({}_{2}\) plume from moving towards zone Beta.  
The risk of fracturing occurring far from the injection well is low as a result of the pressure reduction caused by gas production in the Troll field. The shallowest portion of zone Alpha, which has the only potential for fracturing, is positioned at a depth of 1240 m. This depth is located beneath the caprock, which possesses a fracture pressure of 187 bar. If a safety factor is used, the maximum pressure allowed at the top of zone Alpha will be 150 bar [49].  
During CO\({}_{2}\) injection in well Alpha, the northernmost spill point is reached first, which is approximately located in a grid with an i-index of 60 (Figure 4). Thus, the i-index of 50 was selected as the saturation constraint to avoid the CO\({}_{2}\) plume from extending beyond the 60 th i-index.  
The injected CO\({}_{2}\) volume can be calculated by multiplying the injection rate and the duration of injection, both of which are optimization variables. Furthermore, saturation and pressure distribution can be rapidly predicted utilizing grid-based SRMs. These indicate that the objective function and constraints can be expressed as follows:  
Maximize \(injected\ volume(Rate,Time)\)  
Rate: [1.8 \(\times\) 10\({}^{6}\)-7.6 \(\times\) 10\({}^{6}\)] Sm\({}^{3}\)/day  
Time: [0.6:600] month  
Subjected to:  
\[Saturation_{i_{1},i_{2},k_{1}}(Rate,Time)-0.1\leq 0\]  
\[Pressure_{k_{2}}(Rate,Time)-150\leq 0 \tag{4}\]  
Figure 8: Saturation distribution in 2072. 35-year CO\({}_{2}\) injection in newly designed injection well Alpha.  
where \(i_{1},j_{1}\), and \(k_{1}\) are cell indexes that address the spill points locations, and \(k_{2}\) defines the top of zone Alpha. The minimum injection rate was considered 2 \(\times\) 10\({}^{6}\) Sm\({}^{3}\)/day, which can be increased up to 7.6 \(\times\) 10\({}^{6}\) Sm\({}^{3}\)/day without exceeding the bottom hole pressure limit. Time is the injection period, which is between 25 and 50 years, with 6-month time steps.  
This research used a GA as the optimization technique due to its simplicity of implementation, effective constraint handling, and strong convergence capabilities, imported from the Pymoo package [68] in Python 3.10.0, without any hyperparameter tuning during sampling, crossover, and mutation. The requirement for the time step to be an integer necessitates the selection of an integer value for the CO\({}_{2}\) injection rate within the specified range, which, in turn, enhances the efficiency of the optimization process. Consequently, an integer simulated binary crossover and polynomial mutation are employed subsequent to an integer random sampling. The crossover and mutation functions utilize a probability of 1.0, which means all variables are involved in crossover and mutation, and an eta value of 3, which controls the distribution of new solutions around parental solutions. The default solution provided by the module for handling constraints is the death penalty. The population size is fixed at 100 individuals, and the optimization process will conclude after replacing 100 generations.