# Mitigation of Gravity Segregation by Foam to Enhance Sweep Efficiency
## 4 Gravity Segregation in Layered Model
### Effect of Layer Thickness with Fixed Injection Rate  
We define a dimensionless parameter \(\Omega\) as  
\[\Omega=\frac{L_{g,r}-L_{g,H}}{L_{g,L}-L_{g,H}}. \tag{6}\]  
Here, the segregation lengths in the two-layer model, higher-permeability homogeneous model, and lower-permeability homogeneous model are denoted as \(L_{g,r}\), \(L_{g,H}\), and \(L_{g,L}\), respectively. The homogeneous model (i.e., single layer) upholds equivalent conditions as the two-layer model, except for the variation in permeability. The parameter represented by the symbol \(\Omega\) is utilized to evaluate the influence of the low-permeability layer on the final extent of segregation in a dual-layered system. When the value of \(\Omega\) approaches 1.0 in the two-layer model, it indicates that the low-permeability layer dominates the segregation length. Conversely, a value close to 0.0 signifies the dominance of the high-permeability layer in determining the segregation length. The thickness ratio, i.e., the quotient of the top layer thickness (\(H_{top}\)) to that of the bottom layer thickness (\(H_{bottom}\)), is also considered to characterize the flow behavior. The evolution of segregation length under different thickness ratios and various permeability is illustrated in Figure 8. As the thickness ratio increases, it can be observed that the parameter \(\Omega\) decreases in the case where the lower-permeability layer is situated at the bottom. In situations where the ratio is small, the low-permeability layer situated at the bottom predominates, whereas \(\Omega\) remains constant and approaches 0 as the thickness ratio exceeds a specific value, as illustrated in Figure 8a. In our study, it was observed that for a thickness ratio equal to or exceeding 0.5, the segregation length is predominantly determined by the upper layer with higher permeability. Regardless of the contrast in permeability, the trend observed in \(\Omega\) remains almost constant when the layer with higher permeability is situated at the top. By contrast, the value of \(\Omega\) exhibits a positive correlation with the thickness ratio and attains 1.0 when the layer of lower permeability is positioned at the top, as depicted in Figure 8b. Nonetheless, there exist ubiquitous findings across diverse scenarios. In cases characterized by a greater permeability contrast (i.e., smaller ratio), there is an increase in the requisite thickness ratio for the lower-permeability layer to assert its dominance, thereby magnifying the impact of the higher-permeability layer situated beneath. Conversely, if the higher-permeability layer is positioned at the top, \(\Omega\) exhibits deviating from 1.0 (Figure 8a), which suggests that the presence of the thin layer with high permeability at the top has an impact on the ultimate segregation of the thick layer with low permeability. Similarly, in cases where the layer with lower permeability is situated at the topmost position, the value of \(\Omega\) does not equate to zero (Figure 8b). The observed increase in the ultimate segregation length aligns with the inference that a low-permeability stratum at the uppermost region of the reservoir can effectively bolster sweep efficiency. (i.e., resulting in a greater segregation distance) [15; 19]. Figure 9 shows the average injection rate (i.e., injection rate per meter in each layer) in each case. The average injection rate in each configuration is considerably different. With a bigger permeability contrast, the average injection rate into the high- and low-permeability layer differs significantly: for instance, for permeabilities \(K_{1}\) and \(K_{4}\). Such significant differences affect the segregation process in different layers. If the permeability contrast is mild, the average injection rate is similar in different layers. The final segregation length follows similar trends whether the high- or low-permeability layer lies on the top (see Figure 8 red curves). Figure 9 also demonstrates that the cause of \(\Omega<1\) (Figure 8a) and \(>\)0 (Figure 8b) when a thinner layer is on the top.