Geofluids

Volume 2017 (2017), Article ID 3602593, 12 pages

https://doi.org/10.1155/2017/3602593

## Modeling and Analysis of Magnetic Nanoparticles Injection in Water-Oil Two-Phase Flow in Porous Media under Magnetic Field Effect

^{1}College of Engineering, Effat University, Jeddah 21478, Saudi Arabia^{2}Computational Transport Phenomena Laboratory (CTPL), Division of Physical Sciences and Engineering (PSE), King Abdullah University of Science and Technology (KAUST), Thuwal, Jeddah 23955-6900, Saudi Arabia^{3}Mathematics Department, Faculty of Science, Aswan University, Aswan 81528, Egypt^{4}Faculty of Engineering, University of Regina, Regina, SK, Canada

Correspondence should be addressed to Mohamed F. El-Amin

Received 23 February 2017; Revised 20 June 2017; Accepted 26 July 2017; Published 28 August 2017

Academic Editor: Timothy S. Collett

Copyright © 2017 Mohamed F. El-Amin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this paper, the magnetic nanoparticles are injected into a water-oil, two-phase system under the influence of an external permanent magnetic field. We lay down the mathematical model and provide a set of numerical exercises of hypothetical cases to show how an external magnetic field can influence the transport of nanoparticles in the proposed two-phase system in porous media. We treat the water-nanoparticles suspension as a miscible mixture, whereas it is immiscible with the oil phase. The magnetization properties, the density, and the viscosity of the ferrofluids are obtained based on mixture theory relationships. In the mathematical model, the phase pressure contains additional term to account for the extra pressures due to fluid magnetization effect and the magnetostrictive effect. As a proof of concept, the proposed model is applied on a countercurrent imbibition flow system in which both the displacing and the displaced fluids move in opposite directions. Physical variables, including water-nanoparticles suspension saturation, nanoparticles concentration, and pore wall/throat concentrations of deposited nanoparticles, are investigated under the influence of the magnetic field. Two different locations of the magnet are studied numerically, and variations in permeability and porosity are considered.

#### 1. Introduction

Industry is now looking seriously into using nanotechnology as a viable tool to solving new challenges in several fields. In particular, there has been interest among oil and gas production companies to explore using nanotechnology in solving challenges related to unconventional oil and gas reservoirs, such as those found in tight and shale formations [1–7]. Nanotechnology has been used in different areas of the oil and gas industry from exploration, drilling, production to reservoir monitoring, and refining. The conventional Enhanced Oil Recovery (EOR) methods have several problems from high cost to low oil recovery in addition to operations problems especially in thermal and chemical methods. The nature of nanoparticles results in some useful characteristics such as increased surface area, which at the nanoscale size, does matter when it comes to how molecules react to and bond with each other. So, for example, nanoparticles can be used in EOR, because they are small enough to pass through pore throats in typical reservoirs, and they can be retained by the rock. Ju and Fan [8] calibrated a model for nanoparticles transport in two-phase flow in porous media based on the formulation of the colloid model of fine particles transport in two-phase flow in porous media [9]. El-Amin et al. [10–13] have presented modeling and simulations of nanoparticles transport associated with two-phase flow in porous media. On the other hand, experimental studies of using nanoparticles in EOR have been conducted by Suleimanov et al. [14] and Hendraningrat et al. [15].

One of the prospective applications of nanotechnology is nanoferrofluids, as the flow of such fluids can be controlled by the introduction of external magnetic field. This opens the way for various applications from directing flow in reservoir monitoring, diverting flow in acid jobs to control and boost of injection fluid advancement during pressure maintenance to increase oil recovery. The idea of using a strong external magnetic field with large magnetic susceptibility fluid is to mobilize ferrofluid through porous media. Oil recovery can be increased by using nanoparticles with electromagnetic properties (such as iron oxide, , and zinc oxide, ZnO) under waves generated from an electromagnetic source. Both direct and alternating magnetic fields are under investigation; in our case here we will focus on direct magnetic field. The magnetization of the particles and their attraction toward the magnet causes flow of the magnetic particles suspension. The movement of the magnetic nanoparticles under the magnetic field effect is independent of the orientation of the magnet. In the last few years, a number of publications have been considered nanoferrofluids in oil and gas recovery or environmental applications (e.g., [16–23]).

Borglin et al. [24] conducted experiments to measure the magnetic induction, which converted to magnetic field strength, at various distances in a direction aligned with the poles. McCaig and Clegg [25] presented equations which indicate that ferrofluid magnetization is variable at all locations far away from the magnet due to the decreasing magnetic field strength. The gradient of magnetic field strength varies also with distance from the magnet. Moreover, the force on the ferrofluid decreases with distance from a magnet. Therefore, we may conclude that the distance from the magnet is important and should be reconsidered with taking into consideration its location in particular in the core-scale as well as the direction of flow. On the other hand, it is known that, in the presence of an external magnetic field, the nanoparticles in ferrofluid become magnetized and are pulled toward a magnet. In this work, we attempt to discuss the effect of location of the magnet on the transport of the nanoparticles in porous media.

In the current work, we develop a mathematical model to describe the magnetic nanoparticles-water suspension imbibition into an initially oil saturated porous domain under magnetic field effect. The porous medium is considered initially saturated totally with oil except for a residual amount of the other phase. We consider countercurrent imbibition into a small-scale porous core. This countercurrent imbibition refers to the case in which all the porous medium domain boundaries have no flow except one side. Physical variables are investigated under the influences of magnetic field with two different locations of magnet, namely, right and left to the porous core. Numerical experiments for the two cases are performed and results are introduced in graphical representations.

#### 2. Modeling and Mathematical Formulation

Consider suspension of magnetic nanoparticles injected in an isothermal incompressible water-oil two-phase flow under an external magnetic field. In the following, we describe the mathematical modeling of the problem under consideration.

In the following subsections, we firstly introduce the magnetic force and other magnetic modeling. The magnetic body force, which acts as a body force on the nanoparticles suspension per unit volume, appears in the extended Darcy’s law as presented in the second subsection. Other magnetic parameters such as magnetization and magnetic field strength are also presented in the first subsection. Then, the governing flow equations such as momentum (extended Darcy’s law) and mass conservation (saturation equation) are provided in the second subsection. The third subsection is devoted to the modeling of nanoparticles transport which is coupled with the flow equations via velocity and saturation. Finally, the initial and boundary conditions are presented in the last subsection.

##### 2.1. Magnetic Force

The magnetization (polarization) of the nanoparticles suspension interacts with the external permanent magnetic field to produce attractive forces on each particle. The external magnetic force acts as a body force on the nanoparticles suspension per unit volume which can be expressed as [16]where is the magnetic permeability, is the magnetization, and is the magnetic field strength. The magnetization is a function of , approximated by where the parameters and depend on the particular type of the ferromagnetic material. The values of the initial susceptibility and the saturation magnetization are controlled by the parameters and , respectively. The larger leads to a larger initial susceptibility which corresponds to larger particles or agglomeration of particles. The range of may be of order 10^{4}–10^{5}, while the order of may be of order 10^{−6}–10^{−5}. The magnetic field strength in 1D may be written as follows [16]: where is the residual magnetization (in this work, [T]) and is the distance between the poles of the magnet. In Figure 1, , , and are plotted against .