# Using Low Salinity Waterflooding to Improve Oil Recovery in Naturally Fractured Reservoirs
## Appendix A  
In this Appendix, we present the analytical solution for the proposed approach of estimating the weighting factor in the mixing zone of the low salinity waterflooding scheme using 1D model. The model has already been discussed, where the simulation results illustrated by using saturation profiles of both tracers and water. Then, the weighting factor equation has been concluded based on the results of the tracer simulation model.  
Initially, the water front of the 1D model has been calculated analytically using the Buckley-Leverett theory to validate the simulation results. The fractional flow of the injected water can be determined analytically using the following equation [42; 43]:  
\[f_{w}=\frac{1}{1+\frac{k_{r_{0}}}{k_{nr}}\frac{\mu_{w}}{\mu_{o}}} \tag{9}\]  
where the relative permeability ratio in Equation (9) can be replaced with the exponential function of the plotted relationship between the water saturation and relative permeability ratio for the straight-line section, as illustrated in Figure 11A and expressed in Equation (10), where both \(a\) and \(\mathrm{b}\) coefficients can be calculated [43]:  
\[\frac{k_{r_{0}}}{k_{rw}}=ae^{bS_{w}} \tag{10}\]  
Then, Equation (10) is substituted into Equation (9) to obtain:  
\[f_{w}=\frac{1}{1+ae^{bS_{w}}\frac{\mu_{w}}{\mu_{o}}} \tag{11}\]  
A tangent line was plotted from the initial water saturation to the maximum point in the water cut (\(\mathrm{f_{w}}\)) curve, as demonstrated in Figure 11B to calculate the saturation of the leading waterfront, as well as to determine the distance from the injector at the required time. An example is given in Figure 11C to illustrate the water saturation, the leading front, and the distance from the injector after 462 days. Moreover, this analytical solution was compared with the 1D tracer simulation scenario, which is illustrated in Figure 2. The analytical solution had a sharp waterfront at a distance of 260 ft from the injector. Meanwhile, the result of the numerical simulation exhibited a gradual mixing in the waterfront with an average equal or slightly higher than the analytical solution frontal distance. The distance from the injection point can be estimated using Equation (12), where the (\(\mathrm{df_{w}}\)/\(\mathrm{dS_{w}}\)) can be calculated using the derivative form of Equation (11).  
\[D=\ 5.615\ \frac{W_{i}\ t}{\varphi\ A}\ \frac{df_{w}}{dS_{w}} \tag{12}\]  
[PERSON] [24] has illustrated that the high salinity water moving ahead to form the waterfront followed by low salinity water. The conclusion is the same even when the injected water has the same salinity as the formation of water, as illustrated in Figure 11D. In this example, both injected water and formation water have the same salinity while a tracer concentration of 3000 has used for the injected water compared with 100,000 for formation water. The simulation results demonstrated that a mixing zone of the tracer has formed, which starts with the maximum concentration in the waterfront and ends with the minimum concentration at a closer distance to the injection point. This conclusion is further illustrated using an overlapping plot of the subfigures shown in Figure 11C,D, as shown in Figure 11. In this plot, the leading frontal (i.e., beginning of the mixing zone) located at 305 ft from the injector and ending at 40 ft. Moreover, the mixing region of the brine mixture was situated within a 265 ft interval, which begins with the \(\mathrm{SC_{LoSal}}\) and ends with the \(\mathrm{SC_{Mexeff}}\) to the right. Therefore, the width of the mixing interval that starts altering the reservoir wettability and reduces the