# Determination of Optimum Frequency for Electromagnetic-Assisted Nanofluid Core Flooding
## 3 Results and Discussion
### EM Wave Propagation  
#### 3.4.1 Antenna Impedance and Return Loss  
The simulated input impedance at the lumped port of the antenna is presented in Figure 7 for different operating frequencies ranging from 100 MHz to 2.0 GHz. For an antenna to propagate and deliver maximum power, the input impedance is expected to match with the characteristic impedance of the transmission line according to \(Z_{L}=Z_{A}\), where \(Z_{L}=R_{L}+jX_{L}\) is the transmission line characteristic impedance (composed entirely of reactance (real) part R\({}_{\rm L}\) = 50 \(\Omega\) and capacitance (imaginary) part \(X_{\rm L}\) = 0) and \(Z_{A}=R_{A}+jX_{A}\) is the antenna input impedance. Therefore, the simulated antenna will resonate (propagate maximum power) at a frequency where the imaginary impedance is closest to zero, with a corresponding real impedance close to 50 \(\Omega\). It was observed that the real part of the input impedance of the antenna approaches 50 \(\Omega\) at 0.2 GHz (56.54 \(\Omega\)), 1.4 GHz (51.22 \(\Omega\)), 1.5 GHz (56.01 \(\Omega\)), 1.7 GHz (51.39 \(\Omega\)) and 1.8 GHz (42.62 \(\Omega\)). Nonetheless, the antenna cannot resonate at these frequencies due to the large respective imaginary parts, 111.79 \(\Omega\), 10.82 \(\Omega\), 16.33 \(\Omega\) and 35.85 \(\Omega\). The real and imaginary parts of the antenna input impedance in sandstone saturated with oil, brine and \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Formation** & **Dielectric** & **Conductivity** & **Magnetization** & **Coercivity (G)** & **Retentivity** \\  & **Permitivity** & **(\(\times 10^{-2}\) S/m)** & **(\(\times 10^{-3}\) emu/g)** & **Coercivity (G)** & **(\(\times 10^{-9}\) emu/g)** \\ \hline Dry Sandstone & 4.0 & 3.276 & 10.361 & 47.82 & 91.89 \\ Sandstone + Oil & 6.1 & 2.396 & 4.367 & 37.59 & 29.43 \\ Sandstone + Oil + Brine & 8.1 & 3.487 & 6.221 & 23.91 & 32.05 \\ Sandstone + Oil + Brine & 19.1 & 7.732 & 10.601 & 31.66 & 24.30 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Electromagnetic properties of sandstone samples at different fluid saturations. nanofluid are presented in Figure 7c, as this is of interest for the core flooding process. The imaginary part is noticed to approach zero at 0.8 GHz (3.82 \(\Omega\)) and 1.3 GHz (8.96 \(\Omega\)); this indicates that power stored in the near field (non-radiated power) is minimal at these frequencies. Likewise, the real part of the impedance at these frequencies are 40.51 \(\Omega\) and 46.35 \(\Omega\), respectively; hence the antenna propagates the maximum input power at 800 MHz and 1.3 GHz. Figure 8 shows the simulated return loss of the antenna in four different media, dry sandstone, sandstone saturated with oil, sandstone saturated with oil and brine and sandstone saturated with oil, brine and nanofluid. The essence of analyzing the reflection loss peaks is to determine frequencies at which the least amount of input signal is reflected by the antenna in order to maximize energy delivery to reservoir materials for stimulation and recovery of trapped oil. The antenna has return loss \(<-10\) dB in the range; between 1.09 GHz and extends over 2 GHz (0.91 GHz bandwidth), between 990 MHz and 2 GHz (1.01 GHz bandwidth), between 900 MHz and 2 GHz (1.1 GHz bandwidth), and between 580 MHz and 1.68 GHz (1.1 GHz bandwidth) within the aforementioned four scenarios, respectively. The curve indicates that the antenna has a single reflection loss peak with a corresponding value of \(-20.708\) dB at 1.3 GHz and \(-20.575\) dB at 1.2 GHz for the dry sample and when saturated with oil, respectively. However, the antenna exhibits double reflection loss peaks in the sandstone containing oil and brine (\(-21.12\) dB at 1.0 GHz and \(-21.37\) dB at 1.4 GHz) and that containing oil, brine and magnetite nanofluid (\(-18.91\) dB at 800 MHz and \(-20.00\) dB at 1.3 GHz). The double reflection loss peaks of the antenna within these two media could be attributed to the electrical length of the antenna.