# Mitigation of Gravity Segregation by Foam to Enhance Sweep Efficiency
## 3 Character of Gravity Segregation in Homogeneous Porous Media
### Role of Transition Zone  
In order to examine the underlying reasons for the transition zones depicted in Figure 4, a series of small 2D simulations were carried out to analyze the influence of grid resolution. To achieve higher resolution, we used a domain size of 60 m \(\times\) 1 m \(\times\) 20 m without foam injection, and varied the grid block sizes. The grid block sizes in this case are 0.1 m \(\times\) 1.0 m \(\times\) 0.1 m, 0.5 m \(\times\) 1.0 m \(\times\) 0.5 m, and 1.0 m \(\times\) 1.0 m \(\times\) 1.0 m, respectively. When foam was present, we used a domain size of 300 m \(\times\) 1 m \(\times\) 30 m. Based on our findings, it can be inferred that the introduction of a surfactant leads to a negligible distinction between the transition area connecting the mixed and underride zones. As such, the investigation solely involves altering the vertical grid block size (\(dz\)) to analyze the impact of grid size on the transition zone located beneath the override zone when foam is present. The injected \(f_{g}\) of 75% remained constant across all experimental cases. In accordance with the information presented in Table 1, the remaining parameters remain constant. According to [16], the model offers a technique for calculating the steady-state thickness of the over-/underride zone. This calculation is expressed as follows:  
\[\frac{H_{w}}{H_{g}}=WAG\frac{\lambda_{gg}}{\lambda_{ww}}=\frac{Q_{w}}{Q_{g}} \frac{\lambda_{gg}}{\lambda_{ww}}, \tag{5}\]  
where \(H_{w}\) and \(H_{g}\) represent the thickness of the underride zone and override zone, respectively. Additionally, \(\lambda_{gg}\) and \(\lambda_{ww}\) denote the relative mobility of gas in the override zone and the relative mobility of water in the underride zone, respectively. According to [16], the parameter \(\lambda_{gg}\) can be understood as the gas mobility when irreducible water saturation is reached, while \(\lambda_{ww}\) represents the mobility of water at a saturation level of 100%. The determination of the water-alternating-gas (WAG) ratio entails the consideration of the volumetric injection rates of water (\(Q_{w}\)) and gas (\(Q_{g}\)). Gas saturation profiles at steady-state for different grid sizes in the absence of surfactant are depicted in Figure 6. The segregation length across different cases varies because of the change in grid resolution. The interface between the mixed and underride zones creates a region of transition where the presence of gas is not anticipated in the model under consideration. Based on Equation (5), the height of the override zone is estimated to be approximately 1.32 m. The analytical and numerical results for the segregation point and override zone thickness differ significantly for a grid block size of \(dz=1.0\) m. These differences decrease with a finer grid resolution, as demonstrated in Figure 6a for a grid block size of 0.1 m. The numerical dispersion effect is responsible for the transition zones when the surfactant is absent. Despite the insignificant disparity in total relative mobility behind the shock front, numerical simulations inadequately capture shocks. Increasing the grid resolution decreases the deviation from analytical results, but it comes at a higher computational cost. \begin{table}
\begin{tabular}{c c c c} \hline
**Case** & **Model (Lower), Bar** & **Model (Upper), Bar** & **Simulation, Bar** \\ \hline K = 32.8 mD & 204.0 & 308.4 & 238.4 \\ K = 56.8 mD & 192.9 & 259.8 & 212.5 \\ K = 169.8 mD & 154.3 & 179.6 & 168.2 \\ K = 551.5 mD & 141.7 & 169.8 & 160.8 \\ \hline \end{tabular}
\end{table} 
Table 2: An analysis of injection pressure variations across diverse models. Figure 7 depicts the gas saturation profile (with surfactant present) at steady state under varying grid sizes. Notably, the introduction of foam into the formation significantly reduces gas mobility, resulting in a negligible transition region between the mixed and underride zones. With foam injection, every grid block traverses a saturation range from the injection condition (shock) to the initial condition.