# Determination of Optimum Frequency for Electromagnetic-Assisted Nanofluid Core Flooding
## 2 Materials and Methods
### Reservoir Model Geometry and Electromagnetic Propagation Simulation  
EM propagation of a 17-mm long half-wave dipole antenna in a reservoir environment was simulated based on the earlier measured EM properties (\(\varepsilon\), \(\sigma\) and \(\upmu\)) using COMSOL Multiphysics software. Figure 2 shows the schematic representation of the conceptual geometrical model deployed in the simulation study, where the transmitter was positioned at the injection well.  
Figure 1: Berea sandstone samples (**a**) dry sample, (**b**) sample saturated with crude oil and (**c**) sample saturated with crude oil and magnetite nanofluid.  
A user-controlled mesh option was utilized for the numerical discretization to create a mesh sequence with highly refined elements on the antenna structure and at the edges of the geometry (as shown in Figure 3). Since this is an electromagnetic problem, the maximum mesh element size \(S_{max}\) follows the Nyquist criterion \(S_{max}<\frac{c}{2f\sqrt{\frac{c^{\prime}}{\mu^{\prime}}}}\)[35]. A combination of tetrahedral and rectangular mesh elements was used, and the mesh was divided into three sequences, the first of which, representing the sandstone region, has a maximum element size and growth rate of 1.342 cm and 1.5, respectively. The second mesh sequence used for the antenna structure has a \(S_{max}\) of 0.01 cm, a maximum element growth rate of 1.5 and a curvature factor of 0.6. The maximum element size of the third sequence (the lumped port) is \(1.25\times 10^{-3}\) cm.  
The perfectly matched layer (PML) boundary condition was used to eliminate the boundary reflection phenomenon for computational convenience. The PML is a domain with an anisotropic and complex-valued permittivity and permeability used for encapsulating a geometric domain of interest for the frequency-domain solution of Maxwell's equations. Our choice of a polynomial stretched-coordinate perfectly matched layer (SC-PML) boundary condition that absorbs omnidirectionally was inspired by the report of [36]. The details of the geometric dimensions used for building the model depicted in Figure 3 are given in Table 1. Wave excitation was produced by applying voltage difference at the lumped port (the gap between the two antenna arm lengths) with a coaxial cable transmission line with a characteristic impedance \(Z_{\text{L}}\) of 50 \(\Omega\). The electric field distribution at different distances from the antenna ensuing from the EM radiation was obtained by solving the wave equation in lossy media [37]. Accordingly, the three-dimensional far-field distribution was obtained.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline Reservoir model length (cm) & 30.5 \\ Reservoir model diameter (cm) & 4.0 \\ Transmitter distance (cm) & 1.0 \\ Antenna arm length (mm) & 17.0 \\ Antenna radius (mm) & 1.0 \\ Gap (mm) & 1.0 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Geometric model parameters.  
Figure 3: Antenna and sandstone core geometry mesh.  
Figure 2: Schematic diagram of reservoir geometry with electromagnetic (EM) simulation.