## 5 Discussion  
To facilitate a broad discussion on the modifications and the results presented so far, a brief synopsis of what has transpired thus far needs to be presented. The objective of this work is to apply a BO framework to optimize well placement in a CCS project. Rather than applying BO in a \"out of the shelf\" straightforward fashion, several key modifications and analyses outlined throughout Section 3 were employed, aiming at enhancing optimization efficiency and effectiveness. Firstly, by utilizing a formation with complex geometry rather than a simple \"sugar box\", the issue of infeasible areas within the input space emerged and had to be addressed by implementing a transformation technique. This step allowed us to circumvent infeasible regions that took up the majority of the original input space. This modification was not tax-free, as it introduced a severe issue to the kernel. The non-homeomorphic nature of the map implied that kernel closeness of points in the new modified input space did not translate to the same closeness on the original, thus violating the need for the kernel function to calculate the covariance properly. To address this, modifications were further introduced to the Matern kernel, utilizing inverse mapping to capture real distances between well positions. By recalculating covariances based on actual distances, this adapted kernel accurately represented the spatial relationships crucial for optimal well placement. A significant advancement in the proposed methodology was the integration of our knowledge of permutation invariance into the kernel function. From a mathematical point of view, this innovation enabled the kernel to operate on sets rather than conventional vectors, effectively considering the minimum distances between well positions under permutations. The resulting kernel showed to be instrumental in improving optimization outcomes, as demonstrated through rigorous experimentation in Section 4.2, where BO was run with and without the proposed modifications, with the results being significantly better in the modified case. Clearly, the modified kernel exhibited superior performance, highlighting its pivotal role in driving significant improvements in well placement optimization outcomes. It is our strong belief that a solid foundation has been laid for future research endeavors aimed at optimizing the well placement problem with enhanced precision and efficiency. The proposed method is generalizable to other applications of well placement optimization, not necessarily restricted to a BO framework as long as wells operate under group control, making the input space invariant under permutations. These modifications can also find applications in other learning algorithms characterized by permutation invariant inputs. In addition to the comparisons presented thus far, it is important to address the optimal placement identified by the algorithms and consider what insights engineers' intuition could offer. Generally, in a closed system, producers and injectors should be positioned as far apart as possible. For this aquifer, an alternative configuration could involve injecting  
Figure 10: Maximum obtained CO\({}_{2}\) volume values vs. iteration number. CO\({}_{2}\) at the crest and placing the producers around the periphery, which would potentially enforce the resulting plume to remain relatively stationary and not descend as rapidly. As depicted in Figure 11, the solution derived by the BO algorithm reflects a blend of both engineering perspectives and considerations. Injectors are strategically positioned as far away as possible from the majority of the producers, situated in the region resembling a fork at the base of the aquifer. Conversely, most producers are sited on the opposing side of the crest that resembles a \"hill\". CO\({}_{2}\) naturally migrates upwards, facilitated by a clear pathway to the crest.