# Mitigation of Gravity Segregation by Foam to Enhance Sweep Efficiency
## 2 Problem Description  
Ref. [15] demonstrated that vertical flow barriers, for instance, low-permeability zones, can result in increased recovery compared to a uniform reservoir. However, the impact of barriers or preferential-flow channels on gravity segregation during foam injection remains unknown. This study aims to examine how reservoir heterogeneity affects gravity segregation in foam-application processes in a two-layer reservoir, as depicted in Figure 2, by varying layer thicknesses (\(H_{1}\) or \(H_{2}\)), permeabilities (\(K_{1}\) or \(K_{2}\)), and foam parameters (\(fmmob\), \(fmdry\), \(epdry\)). This model originates from a realistic layered reservoir with a thickness of around 30 m. The permeability ratios are changed, resulting in different thicknesses for each layer, depending on the thickness ratio \(R\). The foam parameters for different rock types are determined based on experimental data fitting.  
The foam parameters for different rock permeabilities were obtained from a study by [27], wherein the influence of rock permeability on the strength of CO\({}_{2}\) foam was examined. The permeability range investigated in their study varies from 32.8 mD to 551.5 mD. The characteristics of the implicit-texture foam model employed in this investigation are comprehensively described in detail in Appendix A. The foam-quality scans were fitted using a least-squares method [28], as shown in Table 1. To simplify the problem, we made the assumption that the wettability of rocks remains unaffected by permeability, alongside postulating that the Corey exponents and residual saturations in water-gas relative permeability exhibit uniformity across all formations. With these assumptions, the foam parameters for the implicit-texture model were derived.  
Figure 3 illustrates the correlation between the gas apparent viscosity and foam quality, pertaining to rocks characterized by diverse permeabilities. The plot exposes two distinct regimes for all permeabilities. Within the low-quality range, the apparent viscosity of foam experiences a progressive increment and eventually attains a peak value as the foam quality increases. However, within the high-quality regime, the apparent viscosity of the foam exhibits a declining trend with the progressive increase in foam quality. Additionally, the results indicate that foam apparent viscosity rises with increasing rock permeability, signifying that foam is stronger in higher-permeability rocks.  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline  & **Permeability, mD** & \(k_{rw}^{0}\) & \(n_{w}\) & \(k_{rg}^{0}\) & \(n_{g}\) & \(S_{wc}\) & \(S_{gr}\) & \(fmmob\) & \(epdry\) & \(fmdry\) \\ \hline \(K_{1}\) & 32.8 & & & & & & & & \(1.02\times 10^{3}\) & \(2.50\times 10^{4}\) & 0.185 \\ \(K_{2}\) & 56.8 & & & & & & & \(1.58\times 10^{3}\) & \(9.40\times 10^{3}\) & 0.171 \\ \(K_{3}\) & 169.8 & & & & & & & & \(3.14\times 10^{3}\) & \(6.96\times 10^{3}\) & 0.155 \\ \(K_{4}\) & 551.5 & & & & & & & & \(9.74\times 10^{3}\) & \(4.76\times 10^{3}\) & 0.136 \\ \hline \end{tabular}
\end{table}
Table 1: Relative-permeability and foam parameters used in this work.  
Figure 2: Schematic two-layer model used in this work.