## 4 Geomechanical Modeling of CO2 Injection  
As emphasized in the Introduction, the modeling of geomechanics for CO2 injection in sandstone and shale formations is a critical aspect of ensuring the safety and efficacy of GCS projects. CMG-EM's Finite Element geomechanical coupling provides a sophisticated tool for analyzing the interactions between rock formations and the injected CO2 [63]. This module relies on the fundamental principles of rock mechanics, including stress-strain relationships, deformation, and failure theories. The software employs various constitutive models, such as elasticity, elastoplasticity, and caprock models, to represent the complex behavior of geological materials under different loading conditions. These models are essential for understanding how reservoir rocks react to the injection of CO2, including potential fracturing and changes in permeability and porosity. The CMG-GEM geomechanics module leverages three fundamental equations to simulate the mechanical response of rock formations to CO2 injection. The equilibrium Equation (7) asserts the force balance within the reservoir, indicating that the divergence of the stress tensor (\(\sigma\)) is balanced by body forces (\(B\)). The strain-displacement Equation (8) relates the deformation of the reservoir to displacement fields (\(u\)), which is essential for understanding how the rock fabric responds to changes. Finally, the stress-strain Equation (9) ties the strain (\(\epsilon\)) within the rock to the induced stress, incorporating the impact of pore pressure (\(p\)), temperature change (\(\Delta T\)), and rock stiffness, embodied in the stiffness tensor (\(K\)) [63]. These core relationships are integral to predicting the deformation of the rock, its fracturing potential, and changes in permeability and porosity, all critical for assessing the effects of CO2 injection. \[\
abla\cdot\sigma-B=0 \tag{4}\]  
\[\epsilon=\frac{1}{2}\Big{(}\
abla u+(\
abla u)^{\mathrm{T}}\Big{)} \tag{5}\]  
\[\sigma=K:\epsilon+(\alpha p+\eta\Delta T)I \tag{6}\]  
where  
\(\sigma\) represents the stress tensor;  
\(B\) denotes body forces;  
\(\epsilon\) is the strain tensor;  
\(u\) is the displacement vector;  
\(K\) is the stiffness tensor;  
\(\alpha\) is the Biot coefficient;  
\(p\) stands for pore pressure;  
\(\eta\) is the thermoelastic coefficient;  
\(\Delta T\) represents the temperature change;  
\(I\) is the identity tensor. In this study, linear elastic modeling was adopted in our flow-geomechanics simulations of ground deformation due to CO2 injection. This conservative approach aligns with our focus on understanding ground deformation. It is worth noting that if plasticity were to occur, it would likely lead to even larger displacements. Therefore, by using linear elasticity, we are taking a cautious approach and providing a baseline for potential deformation. This method strikes a balance between accuracy and computational efficiency, enabling us to realistically represent the mechanical behavior of the reservoir under CO2 injection stress while managing computational resources effectively. Our reservoir model's predictive capability for geomechanical responses to CO2 injection is contingent on an accurate representation of the in situ stress profile. This profile is established by mapping pore pressure and minimum total stress from a depth of \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Sandstone** & **Shale** \\ \hline \(S_{gt}\) & 0.25 & 0.35 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Maximum gas trapped due to hysteresis of relative permeabilities and capillary pressure [65]. \(-2900\) m to \(-3400\) m within the injection well. These mappings are important for evaluating rock deformation potential and failure risk. We assume hydrostatic conditions for the pore pressure gradient, while the minimum stress gradient is aligned with observed values for passive margin regimes, specifically reflecting the low tectonic stress environment of Bra'il's offshore Campos Basin.