## 3 Results and Discussion
### Multi-Objective Optimization with Well Allocations  
Multi-objective optimization requires a lot of simulation runs and, thereby, this work selects two different aquifers based on the DGDA result (Table 2; Figure 4); one is the less heterogeneous (\(L\) aquifer; less heterogeneous relevant to the small value of Dykstra-Parsons coefficient) and the other is highly heterogeneous (\(H\) aquifer; the high heterogeneity). Figure 4 demonstrates spatial distributions of the key properties influencing the trapping amount: Figure 4a,b show permeability and porosity of \(L\) aquifer, while Figure 4c,d illustrate those of \(H\) aquifer. Upon comparison of the two aquifers, \(L\) aquifer is capable of storing more CO\({}_{2}\) and induces slow fluid transport through sandstones (because \(L\) aquifer has the smaller sandstone permeability and the larger sandstone porosity; refer to Table 2). On the other hand, \(H\) aquifer is more ambiguous in terms of fluid transport. It has a large portion of shale formations (negative to the continuity of fluid flow) but more permeable sandstones (favorable to a transporting velocity). The former, i.e., shale beds, would be advantageous for long-term CO\({}_{2}\) geological storage, while a drawback would be having to maintain the lower injection pressure since shale can increase the operating \begin{table}
\begin{tabular}{c c c} \hline \hline
**Property (Abbreviation) 1** & **\(L\) Aquifer** & **H Aquifer** \\ \hline PoroSand & 0.274 & 0.220 \\ PermSand & 301.5 & 448.7 \\ SVR & 2 & 20 \\ VDP & 0.3488 & 0.9169 \\ StdPoro & 0.012 & 0.012 \\ StdPerm & 14.0 & 30.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Two heterogeneous aquifers for multi-objective problem with well allocation. Figure 4: Heterogeneous aquifer models to carry out multi-objective optimization with well allocation: (**a**) permeability distribution of \(L\) aquifer model with relatively low heterogeneity; (**b**) porosity distribution of \(L\) aquifer model (the large pore volume); (**c**) permeability distribution of \(H\) aquifer model with high heterogeneity; (**d**) porosity distribution of \(H\) aquifer model. (**a,b**) are for \(L\) aquifer model. (**c,d**) are for \(H\) aquifer model. The distributed permeability is absolute permeability to \(x\) direction. cost to inject CO\({}_{2}\) continuously. The more permeable sandstone has double-sidedness as well; the higher transmissivity contributes towards the spread of supercritical CO\({}_{2}\) widely, to occupy additional pore spaces, but can increase the risk of leakage. Therefore, this paper selects \(H\) aquifer with overall mean permeability including both sandstone and shale similar to \(L\) aquifer model, notwithstanding that \(H\) aquifer is more heterogeneous; the entire mean permeability of \(L\) aquifer is 227 millidarcy and it is 256 millidarcy for \(H\) aquifer, which occupies less permeable shales and reduces the difference in PermSand from 147.2 millidarcy (=448.7-301.5) to 29 millidarcy (=256-227). Shales distribute sparsely, are intercalated with the massive sandstone, and cannot sequestrate CO\({}_{2}\) plumes completely but delay CO\({}_{2}\) movement. In short, the \(H\) aquifer model has more shale, higher PermSand, and smaller pore volume than the \(L\) aquifer model. The \(H\) aquifer model is more heterogeneous with a higher Dykstra-Parsons coefficient. The multi-objective problem is set to minimize two objectives (Equation (5)): one (\(f_{1}\)) is to minimize the sum of maximum BHPs (that is, injection pressure at the well bottom) assigned at each injector, and the other (\(f_{2}\)) is related to the trapping volume observed after 200 years. The trapping volume (\(V_{\text{Trap}}\)) is the same as the response of DGSA, i.e., the volume of combined structural and residual trapping, to evaluate the actual value of safe sequestration. Because the \(V_{\text{Trap}}\) would be large, \(f_{2}\) is defined as the minus value of the trapping volume divided by the total injection amount. To make a minimizing problem, the minus sign is added.