# Determination of Optimum Frequency for Electromagnetic-Assisted Nanofluid Core Flooding
## 1 Introduction  
The electromagnetic (EM) assisted enhanced oil recovery (EOR) method has gained significant attention for the recovery of residual oil [1]. The transfer of EM energy to reservoir materials depends on the molecular composition, such as the dipolar and magnetic molecules having free or bound charges like electrons or ions [2; 3; 4]. Therefore, EM waves behave differently at different interfaces within the reservoir [5; 6]. The propagating EM wave exerts forces on the molecules, which then align in a direction parallel to the external EM field [7; 8; 9]. Application of electromagnetic (EM) waves of certain frequency ranges has been used in nanofluid core flooding experiments for stimulating the recovery of residual oil after water flooding [10; 11; 12; 13; 14]. The potential efficiency of this method has been demonstrated by several experimental and numerical studies [8; 15; 16; 17; 18; 19; 20; 21; 22]. Different recovery mechanisms other than the common EM heating method have been reported recently. For instance, the application of a magnetic field gradient for the polarization of ferromagnetic nanofluid during nanoflooding experiments has been demonstrated [3]. A considerable recovery factor attributed to interfacial tension (IFT) reduction [23] and hyperthermia phenomena has also been reported with the application of EM waves (frequency range 500-1000 MHz) [11; 24; 25]. There have also been similar reports on the use of EM waves for wettability alteration, implying an improvement in reservoir oil production rate [11; 16; 25; 26; 27]. Currently, few research articles have been generated on transmitter design considerations for EM-assisted oil recovery processes. For instance, numericalstudies on EM-assisted oil recovery using a shaped dipole antenna was conducted by [28], likewise, a cylindrical antenna model using the wavelets analysis method has also been reported [29]. Similarly, dimensional constraints have been considered in simulating three types of antennae at different operating frequencies, 915 MHz, 2.45 GHz and 5.8 GHz [4]. Despite the amount of reported studies, the effect of reservoir EM properties at varying formation fluid contents on the efficiency of the EM transmitter, and its propagation pattern have not been considered. Most of the reported studies irradiate the reservoir with EM waves of an arbitrary frequency (sometimes based on dimensional constraints) without considering the medium's EM response, and its effect on the antenna operating frequency. Thus, this area requires further investigation. This is crucial because the operating frequency, penetration depth and radiation pattern of a transmitter depend on reservoir EM properties. These properties include the electrical conductivity \(\sigma\), dielectric permittivity \(\varepsilon\) and magnetic permeability \(\mu\), all of which vary with fluid content, and are frequency dependent in varying degrees [30; 31]. Consequently, it is vital to factor reservoir EM properties into the determination of antenna dimensions and operating frequency to achieve the desired uniform far-field pattern, homogenous EM energy dissipation and to avoid hot spots on the antenna surface [25]. Furthermore, the three-dimensional far-field pattern of a plane wave can be obtained from Equation (1), expressed in terms of the complex permittivity, permeability and conductivity of the medium [5]. \[\
abla\times\mu_{r}^{-1}(\
abla\times\mathbf{E})-k_{0}^{2}\bigg{(}\varepsilon_{r} -\frac{j\sigma}{\omega\varepsilon_{0}}\bigg{)}\mathbf{E}=0 \tag{1}\]  
where \(\mu_{r}\) is the relative permeability, \(\mathbf{E}\) is the electric field vector, \(k_{0}\) is the wavenumber in free space, \(\varepsilon_{r}\) is the relative permittivity, \(\omega\) is the angular frequency, \(\sigma\) is conductivity and \(\varepsilon_{0}\) is the permittivity of free space.