# Mitigation of Gravity Segregation by Foam to Enhance Sweep Efficiency
## 3 Character of Gravity Segregation in Homogeneous Porous Media
### Gas-Water Coinjection and Foam Injection
The mixed zone is identified at a saturation of \(S_{w,mix}\) based on [PERSON]'s model. To mitigate the effects of numerical dispersion, the delineation of the mixed zone is established based on the criterion of water saturation being equivalent to or lower than (\(S_{w,mix}+0.001\)). The segregation length is significantly affected by numerical dispersion for the case with the absence of foam, which is discussed below. Figure 5b depicts the considerable influence of foam quality on the segregation length. This impact is achieved through the modification of the aggregate relative mobility within the mixed zone, as described in Equation (1). In the high-quality regime, a decrease in foam quality results in an increase in the segregation length, whereas in the low-quality regime, the opposite trend is observed. The point of transition between the two regimes, characterized by the peak apparent viscosity of the foam, is accompanied by the attainment of the maximum segregation length value, which corresponds to the lowest total relative mobility. The numerical model employed in our study exhibits a commendable level of concurrence with the analytical solutions proposed by [15; 16], disregarding the numerical dispersion impact. Figure 4: Distribution of water saturation (\(f_{w}=25\%\)) in a homogeneous porous medium under steady state, with the white dashed line indicating the predicted segregation point by Equation (1). Notably, both cases exhibit transition regions characterized by reduced water saturation compared to the initial state. Ref. [20] demonstrated a significant relationship between injection pressure and the phenomenon of gravity segregation within the cylindrical flow. Building upon their findings, the current investigation broadens the scope of this correlation's implementation to encompass a rectangular coordinate system, as outlined in Appendix B. The correlation provides an upper limit on the injection pressure required for the onset of gravity segregation  
\[p(r_{w})-p(L_{g})=\frac{L_{g}^{2}k_{z}(\rho_{w}-\rho_{g})gW}{k_{h}A}, \tag{3}\]  
and the lower bound is  
\[p(r_{w})-p(L_{g})=\frac{L_{g}^{2}k_{z}(\rho_{w}-\rho_{g})gW}{2k_{h}A}, \tag{4}\]  
where \(p(r_{w})\) and \(p(L_{g})\) denote the pressure at the wellbore and at the segregation point, respectively. Equations (3) and (4) do not involve either \(Q\) or \(\lambda_{rt}^{m}\), implying that the extent of the segregation point is exclusively governed by the pressure exerted at the injection well, in agreement with the observations made by [20]. Additionally, the pressure difference between the injection point and the segregation point in the upper bound is twice that in the lower bound. These observations are elaborated in Appendix B. Table 2 presents a comparative analysis between the injection pressure values derived from the model and simulations. The mixed zone's shape is depicted in Figure 3(b), which closely resembles the case where the mixed-zone height decreases proportionately with the total flow rate, corresponding to the lower-bound assumption described in Equation (4). It is noteworthy that the injection pressure obtained from simulations closely approximates the lower threshold. However, it is important to acknowledge that using this approximation, i.e., based on the mixed zone's shape, may lead to deviations while predicting the injection pressure in this study. Therefore, it is essential to develop a model that accurately characterizes the correlation between injection pressure and the morphology of the mixed zone. Furthermore, it should be emphasized that attaining an equivalent segregation point using stronger foam necessitates an elevated injection pressure. Figure 5: An evaluation of the concordance between an analytical model and simulation outcomes. Injection rate is 10 times less for the foam case. (**a**) foam absent; (**b**) foam present. ‘TZ’ is an abbreviation of ‘Transition Zone’.