## 4 Case Study
### Kernel Comparisons  
To further evaluate the performance of the proposed method, a comparison between the modified kernel against the original Matern kernel and an approach that randomly selects well positions is presented. This test is performed on a more strenuous schedule having a higher target injection rate, increased by more than 50%, as seen in Table 4. The target injection rate now corresponds to 3.6 MMtn/yr and the maximum sequestration volume to 142 MMtn.  
Figure 8: “Vector Distances” and “Set Distances” between subsequent inputs for all six investigated models.  
This comparison aims to elucidate the impact of the two modifications made to the kernel described in Section 3 on the final outcome. Specifically, we seek to understand their influence on the end result, as well as their ability to expedite the optimization process. The acquisition function utilized is EI with moderate hyperparameter values, striking a balance between exploration and exploitation. By evaluating these aspects, valuable insights can be gained into the efficacy and efficiency of the proposed modified kernel in comparison with both the original kernel and a random selection strategy.  
As illustrated in Figure 9, the modified kernel surpasses the two alternatives in terms of maximum CO\({}_{2}\) sequestered. Furthermore, it demonstrates consistent improvement, initiating enhancement as early as at the third timestep and continuing to refine its performance a total of four times before the 15 th timestep. In contrast, the original kernel demonstrates improvement only twice, and its performance is comparable only to that of the random process. This stark discrepancy underscores the critical importance of effectively managing kernel functions. Failure to comprehend and address the nuances of optimization caveats would have hindered our ability to achieve such promising outcomes.  
Notably, the original kernel function's performance is akin to that of a random process. This outcome is attributable not only to improper distance calculations based on the 1D modified input space in the covariance but also to the lack of comprehensive information regarding permutation invariance and the need to manage a considerably larger input space as a consequence. Figure 10 further supports this argument by demonstrating that the density distribution of the results of the original kernel and the random optimization  
\begin{table}
\begin{tabular}{l c} \hline \hline
**Parameter** & **Value** \\ \hline (Target) Injection rate (CO\({}_{2}\)) & 170 MMscf/day \\ (Target) Production rate (water) & 77 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\({}_{2}\)) & 2.44 Tscf \\ \hline \hline \end{tabular}
\end{table}
Table 4: Schedule targets and constraints.  
Figure 9: Maximum CO\({}_{2}\) sequestered volume achieved vs. iteration number.  
are very close, as opposed to the modified kernel's results, where the density function is clearly skewed towards the optimal value.