# Mitigation of Gravity Segregation by Foam to Enhance Sweep Efficiency
## Appendix B Correlation between Injection Pressure and Segregation Length  
Ref. [20] provided empirical evidence supporting a relationship between the injection pressure and the manifestation of gravity segregation in the context of cylindrical flow. Their findings suggest that the sole means of effectively controlling gravity segregation during the continuous injection process into a specific reservoir is through the elevation of pressure at the injection well. In this research, we propose an extension of the previously mentioned association within the context of a Cartesian coordinate system. Specifically, we put forward the conjecture that the bottom-hole pressure of the injection well corresponds to the pressure exhibited by the interconnected reservoir block.  
Figure 1 demonstrates two distinct shapes of the mixed, override, and underride zones. In Figure 1, The proportionality between the quantity of water and gas within the mixed zone is contingent upon the ratio between the position denoted by \(x\) and the segregation point represented by \(L_{g}\). Hence, one can deduce that the vertical extent of the mixed zone is dependent on the total rate of fluid flow within the mixed zone. By way of contrast, it can be observed from Figure 1 that the entirety of the gas and water phases persist within the mixed zone until such time that they undergo segregation, leading to a progressive reduction in the mixed zone's height with greater distance from the point of injection, culminating shortly prior to the occurrence of the segregation point. The aforementioned instances depict two distinct thresholds of pressure injection within the water and gas coinjection procedure.  
In the first scenario, the overall horizontal superficial velocity, denoted as \(U_{t}\), within the mixed zone of the linear flow, is expressed as a linear equation of the dimensionless ratio \(x/L_{g}\). This relationship suggests that the overall flow rate within the mixed zone diminishes proportionally with it:  
\[U_{t}=\frac{Q}{A}(1-\frac{x}{L_{g}})=-K_{h}\lambda_{rt}^{m}\frac{dP}{dL}, \tag{10}\]  
where \(K_{h}\) is the horizontal permeability and \(x\) is position. Through the process of integration, extending from the well to the stage of segregation, it is possible to formulate the pressure required for injection:  
\[p(inj)-p(L_{g})=\frac{QL_{g}}{2 AK_{h}\lambda_{rt}^{m}}. \tag{11}\]  
Here, \(p(inj)\) and \(p(L_{g})\) denote the injection pressure and the pressure at the segregation point, respectively. Subsequently, by reordering the formula presented in Equation (1), an equation for the variable \(Q\) can be obtained, which can then be substituted into Equation (11) to obtain the expression  
\[p(inj)-p(L_{g})=\frac{L_{g}^{2}K_{z}(\rho_{w}-\rho_{g})gW}{2 AK_{h}}. \tag{12}\]  
In the second scenario, wherein the mixed zone, the total flow rate is uniform, Equations (10)-(12) become  
\[U_{t}=\frac{Q}{A}=-K_{h}\lambda_{rt}^{m}\frac{dP}{dL}, \tag{13}\]  
\[p(inj)-p(L_{g})=\frac{QL_{g}}{AK_{h}\lambda_{rt}^{m}}, \tag{14}\]  
and  
\[p(inj)-p(L_{g})=\frac{L_{g}^{2}K_{z}(\rho_{w}-\rho_{g})gW}{AK_{h}}. \tag{15}\]  
It should be noted that neither the variable \(Q\) nor the parameter \(\lambda_{rt}^{m}\) is incorporated into Equation (12) or Equation (15).