# Determination of Optimum Frequency for Electromagnetic-Assisted Nanofluid Core Flooding
## 3 Results and Discussion
### Reservoir Rock Electromagnetic Properties  
Figure 6a shows the variation of relative dielectric permittivity \(\epsilon^{\prime}\) of the sandstone sample under the different possible scenarios encountered in nanofluid core flooding experiments. It is apparent from the figure that dielectric dispersion generally dominates the lower frequency portion (from 1 MHz to 1.5 GHz) of our measurement range, after which the frequency dependence of \(\epsilon^{\prime}\) becomes weaker and attains quasi-stable values. Since relative permittivity is a measure of the degree of polarization occurring in a material [43], the various exhibited peaks at different frequencies could be attributed to specific polarization mechanisms [41], of which dipolar mechanisms are dominant in this case. It is noteworthy to point out that the \(\epsilon^{\prime}\) of the sample not only increases with the inclusion of brine and nanofluid, but dispersion peaks also shift to a lower frequency. Figure 5: FESEM results showing **(a)** pore size distribution of the dry sandstone sample, **(b)** EDS analysis of element distribution of dry sandstone, **(c)** an image of a pore throat mapped after saturating with oil, brine and Fe\({}_{3}\)O\({}_{4}\) nanofluid and **(d)** EDS analysis of elemental distribution confirming the presence of the injected oil and nanofluid. Figure 6: **(a) Dielectric permittivity of the sandstone sample at different saturations and (b) permittivity and conductivity variation of the sample saturated with brine, oil and nanofluid. **The frequency-dependent conductivity \(\sigma_{\rm ac}\) (obtained from the imaginary permittivity according to Equation (4)) and \(\varepsilon^{\prime}\) of the sandstone sample saturated with 40% oil, 60% brine and nanofluid is shown in Figure 6b. The frequency-dependent conductivity of the medium displays an increasing trend with frequency as opposed to the decreasing value of \(\varepsilon^{\prime}\). Moreover, the effective conductivity \(\sigma_{\rm eff}\) of the reservoir medium is composed of the ohmic \(\sigma_{\rm ohmic}\), and the dielectric loss part \(\sigma_{\rm ac}\) (the frequency-dependent) as \(\sigma_{eff}=\sigma_{\rm ohmic}+\sigma_{\rm ac}\). We therefore conclude from these results that the irradiated EM wave should be varied within a frequency range less than 2 GHz. This will enable effective propagation of the EM wave, as attenuation of the EM wave will increase at a higher frequency (as indicated by Equation (2)) leading to a reduction of the penetration depth \(\delta\) due to increasing conductivity at a higher frequency, where \(\varepsilon^{\prime}\) is almost constant. Figure 6a,b clearly indicates that approximately 2 GHz is the upper bound of frequency for an effective response of the reservoir materials to the irradiated EM wave within the tested samples. This is in agreement with the report of [PERSON] [44], who placed the lower frequency limit at 300 MHz and the upper bound at 1-2 GHz for effective EM propagation in the reservoir medium. Therefore, our subsequent investigation in this work was restricted to the frequency limit of 100 MHz to 2 GHz. It can be concluded from the results that, while the injection of Fe\({}_{3}\)O\({}_{4}\) nanofluid increased the overall EM response of the reservoir, another important point is the reduction of the required EM frequency for the EM-assisted nanof flooding experiment, hence increasing the EM penetration depth, polarization effect and improved activation of the injected nanofluid for stimulation of the residual oil. The sandstone sample shows a weak magnetic property (Ms of 10.361 memu/g), which conforms with earlier reports in the literature that most reservoir rocks do not possess intrinsic magnetic properties and are therefore described electrically by the conductive and capacitive properties [45]. However, a variation of the saturation magnetization could be noticed with different fluid contents. Thus, the EM response of the medium entirely depends on the volume fraction of the injected nanofluid.