## 5 Methodology and Results
### CO\({}_{2}\) Injection Flow Results
Panels (**B**,**C**) depict vertical cross-sections at 20 and 70 years, respectively, highlighting pressure dissipation into the surrounding shaly rocks over time. On panels (**A**,**B**), the reservoir is depicted in white. which accommodate and redistribute the excess pressure over the long term, as observed by other authors in similar analyses regarding the caprock's integrity [83, 84]. In Figure 7, the distribution of the CO\({}_{2}\) molar fraction within the reservoir and its immediate vicinity is illustrated at two distinct time points: at the end of the injection period and after an extended monitoring phase. Figure 7A shows a 3D view at the 20-year mark, with the CO\({}_{2}\) molar fraction concentrated around the injector, where red denotes the highest concentration. The presence of CO\({}_{2}\) within the reservoir is contained primarily due to the significant capillary pressure in the shale caprock and its low permeability, which act as barriers to migration. Figure 7B,C depict vertical cross-sections near the injector well at 20 and 70 years, respectively. At the end of the injection phase (Figure 7B), the CO\({}_{2}\) has predominantly remained within the reservoir boundaries. By 70 years (Figure 7C), there is a slight presence of CO\({}_{2}\) detectable in the surrounding rock formation mainly through diffusion or a numerical artifact driven by diffusion due to the course grid considered in our model, as discussed in [59, 85]. However, this penetration is minimal and does not constitute a leakage concern as absence of fractures or other flow localization features in this model is assumed. The limited movement of CO\({}_{2}\) into the surrounding rocks illustrates the combined influence of the caprock's low permeability and the high capillary entry pressure, which together serve as a robust containment mechanism, ensuring the long-term integrity of the geological storage. Understanding the behavior of the CO\({}_{2}\) plume within the geological storage is important for assessing the effectiveness of carbon capture and sequestration strategies. Observing the plume's behavior over time allows us to confirm that the physical characteristics of the reservoir and the injected CO\({}_{2}\) are in agreement with predictive models, ensuring the reliability of long-term storage solutions. In this context, Figure 8 illustrates the CO  
Figure 7: CO\({}_{2}\) global molar fraction post-injection and monitoring phases. Panel (**A**) shows the CO\({}_{2}\) concentration around the injector at the end of the 20-year injection period. Panels (**B**,**C**) reveal vertical cross-sections at 20 and 70 years, respectively, with minimal CO\({}_{2}\) migration into the caprock. On panels (**A**,**B**), the reservoir is depicted in white. migration within the reservoir at two times. The left panel of Figure 8 reveals the state of the reservoir immediately after 20 years of injection, highlighting the buoyant rise of CO\({}_{2}\) towards the less dense upper regions, a behavior that underscores the consistency of our simulation's fluid and rock physics. The right panel demonstrates the situation after 70 years. In this period between 20 and 70 years, the CO\({}_{2}\) within the reservoir has moved upward but not significantly beyond its boundaries, indicating the effectiveness of the caprock's structural integrity and capillary forces in limiting vertical movement. This containment is further evidenced by the minimal lateral dispersion into the surrounding rocks, evidencing the robustness of natural trapping mechanisms and the security of CO\({}_{2}\) storage over an extensive monitoring period. Even using a coarse mesh, the model was able to capture the vertical movement of CO\({}_{2}\), demonstrating the buoyant rise and containment within the geological formation over time. This outcome suggests that even with less detailed spatial resolution, key processes governing CO\({}_{2}\) migration and trapping can be effectively simulated, providing valuable insights into the overall behavior of the CO\({}_{2}\) plume.