## 4 Case Study
### Exploration-Exploitation Trade-Off  
The first phase of testing the BO algorithm involves the utilization of all three candidate acquisition functions introduced in Section 2.3, shown in Equations (11)-(13). Each function is utilized twice, once with explorative hyperparameters values and once with exploitative ones. In total, six BO models are generated, with each one being trained through 50 iterations, by an equivalent amount of simulation runs conducted for each model to refine the surrogate objective function. Details of the selected schedule are outlined in Table 3.  
In the table above, \"M\" denotes 1000, \"MM\" denotes 10\({}^{6}\), and \"T\" corresponds to 10\({}^{12}\) while \"stb\" and \"scf\" represent units of volume, i.e., stock tank barrel and standard cubic feet, respectively, measured under surface conditions. The theoretical maximum volume shown in Table 3 was calculated by multiplying the target injection rate with the duration of the simulation and represents the target value for the optimizer to reach. However, it is crucial to note that this value may not be feasible without violating the imposed constraints; therefore, it may not be achievable under any set of well configurations. Mass-wise, the target injection CO\({}_{2}\) rates correspond to 2.3 MMtn/yr and the maximum sequestration volume to 90 MMtn. At first, the performance of all six models is compared in terms of maximum storage value attained by timestep, as seen in Figure 6. \begin{table}
\begin{tabular}{l c} \hline \hline
**Parameter** & **Value** \\ \hline (Target) Injection rate (CO\({}_{2}\)) & 110 MMscf/day \\ (Target) Production rate (water) & 50 Mstb/day \\ (Constraint) Max production rate (CO\({}_{2}\)) & 5 MMscf/day \\ (Constraint) Max bottomhole pressure & 9000 psi \\ (Constraint) Min bottomhole pressure & 3000 psi \\ \hline Duration & \(\approx\) 40 years \\ Theoretical maximum sequestration volume (CO\({}_{2}\)) & 1.58 Tscf \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Schedule targets and constraints. Figure 6: Maximum obtained CO\({}_{2}\) storage value vs. iteration number. The first component of the series names signifies the hyperparameter value selection guiding the model towards either exploration (XPR) or exploitation (XPT), while the second component indicates the acquisition function employed (see Equations (11)-(13)). It is notable that all models approach a storage result close to the theoretical maximum already by iteration 15. Certain models (XPT EI, XPT PI, and XPR PI) are fortunate to nearly achieve their optimum value early on, possibly due to a lucky initial guess. This suggests that the targeted objectives of the schedule are relatively easy to maintain throughout the simulation. In contrast to other exploitative models, XPT UCB attains an optimal value by the third iteration. Both XPT EI and XPT UCB demonstrate their ability to refine their predictions in later iterations. However, XPT PI peaks at the 10 th iteration but fails to reach the \(1.5\cdot 10^{9}\) mark, likely because PI inherently leans towards exploitation. In contrast, XPR PI continues to exhibit improvements even after 20 iterations. Notably, both XPR UCB and XPR EI consistently find enhancements throughout the iterations, owing to their explorative nature as well as initially exploring sub-optimal regions. Since the results produced for the max value seem to be heavily reliant on the initial estimation, the distribution of the storage results that each model generated at every iteration is investigated, as illustrated in Figure 7, to offer valuable insights into the impact of the strategy on model performance. Notably, exploratory strategies (illustrated in the right-hand-side plots) demonstrate a higher susceptibility to unfavorable outputs, which arises from their emphasis on exploring uncharted territories within the input space.