Analytical solutions of polymer transport in porous media under magnetic effect

The analytical solution is helpful for understanding the mechanism and physical impacts and validating a numerical method through the model issue. This paper is devoted to developing analytical solutions for the problem of magnetic polymer transport in porous media. The mathematical model has been firstly developed then the analytical solutions have been obtained. The magnetization is treated as a nonlinear function of the magnetic field strength. The effects of the parameters of magnetic/polymer/rock on the polymer concentration, pressure and velocity have been investigated. It was found that the magnetization enhances velocity and concentration; and has a significant effect on concentration for high permeabilities. Also, it was found that for high permeabilities the advection dominates, but for lower permeabilities the diffusion dominates. As the adsorption rate increases the polymer solution concentration decreases.


Introduction
Polymers are widely used in many industrial applications, in particular in the chemical industry [1] due to their wide range of characteristics. For example, polymer flooding is considered one of the most effective methods of Enhanced Oil Recovery (EOR). When a polymer is injected with water into an oil reservoir the physical properties of fluids and rocks can be changed. For example, polymer injection reduces the water mobility and thereby improving the volumetric-sweeping effect and the local displacement of the water-flood [2][3][4][5]. In addition, the dissolved polymer molecules increase the brine's viscosity, which in turn increases the water-front saturation. Also, due to the large polymer molecules, the permeability can also be reduced. In order to enable polymer expected role to control the mobility ratio, the polymer is always used with another EOR method like Polymer-Alternating Gas (PAG) method, surfactant-polymer flooding, Alkali-Surfactant-Polymer (ASP) flooding, etc. [6]. In the EOR polymersflood, the Hydrolyzed PolyAcrylaMide (HPAM) has been widely used. However, the produced-water control is one of the common difficulties of using the HPAM for EOR. This is because in re-injected water, the residual HPAM may increase oil content and viscosity.
On the other hand, the magnetic field can be used with polymers in some applications such as the magnetic nanoparticles implementation to eliminate pollutants from the produced water without chemicals [7][8][9][10]. Another example is the usage of magnetic resonance imaging and measurements to monitor oil displacement by water-flooding and polymer flooding [11]. Also, Ming et al. [12] presented a magnetic resonance investigation for low-salinity waterflooding for EOR. An experimental study was presented by Lyu et al. [13] to investigate the application of NMR to spontaneous imbibition recovery of tight sandstones. In order to mobile a magnetic ferrofluid through porous media, a strong external magnetic field may be used. For example, an electromagnetic source, can be used to enhance oil recovery by magnetic ferrofluids [14][15][16][17][18].
A number of publications could be found in using magnetic field with polymers [19], however, there are a very few research on finding analytical solutions of the polymers transport in porous media under the effect of magnetic field. For example, Ford Versypt et al. [20] have derived an analytical solution to a reaction-diffusion model for autocatalytic degradation and erosion in polymer microspheres. Also, Lyu et al. [21] have introduced an analytical solutions to the models of the surface and bulk erosion of solid polymers.
In the present paper, we developed analytical solutions for the model of transport of magnetic polymer solution in porous media under the effect of external magnetic field. The remaining sections of this paper are arranged as; in Section 2, we develop a mathematical model consists of a group of differential equations, algebraic constraints, and initial/boundary conditions. Then, in Section 3, we introduce an analytical solution of the governing equation.
In Section 4, sensitivity analysis and physical discussion are presented. Finally, the conclusion is drawn in Section 5.

Modeling and formulation
In this section, we develop the mathematical model of the problem of a single-phase flow with polymer-water flow under the effect of an external magnetic field (see Fig. 1). The polymer suspension gains properties from both liquid and magnetized solid particles. If the polymer suspension is subjected to an external magnetic field, it flows toward the magnetic field, and the flow resistance increases. In the absence of the magnet, nearby polymer acts as a normal liquid. In the following subsections, the model assumptions are stated, the mathematical models of polymer, flow, magnetic field, polymer viscosity, and initial/boundary conditions are sequentially presented.

Assumptions
The following assumptions are hold for the mathematical model. The flow is to be obeyed Darcy's law and the process is isothermal. The typical polymer size is smaller than the typical pore size. The fluid is compressible, and the rock is compressible and anisotropic. The viscosity of the polymer solution depends on the polymer concentration and the salinity coefficient. The mixture of water and polymer is assumed to be ideal (a homogeneous mixture of substances that has physical properties linearly related to the properties of the pure components). It is also assumed that the medium is free of polymer particles and a permanent magnet is the source of the magnetic field, we can use the simple equation to calculate the magnetic field strength directly. This assumption decouples the calculation of the external magnetic field from the polymer distribution. At a given location the gradient of the magnetic field strength can be calculated by a simple first-order differencing in each of the coordinate directions [22].

Polymer model
The balance equation of the polymer concentration is given by [23,24], The higher-order terms are ignored because dc << 1. The permeability reduction factor is defined by, where RRF is the ratio between the permeability before and after the polymer injection; RRF stands for the "Residual Resistance Factor". For seeking simplification, we consider RRF ¼ 1, i.e. R k ¼ 1, thus we ignore its effect on the transport equation. Therefore, the polymer transport equation (1) can be rewritten in the following form, such that,

Flow model
The external magnetic force acts as a body force of the momentum conservation equation, the 1D Darcy's equation is written as, The continuity equation is, where k [m 2 ] is the permeability, and p [Pa] is the pressure. l e [kg m À1 s À1 ] is the effective viscosity. In the absence of the magnetic field, gðxÞ ¼ 0 [m 2 s À1 ]. Therefore, one obtains, Also, it is important to refer that GðxÞ ¼ R x 0 gðxÞdx and GðxÞj x¼0 ¼ u 0 .

Magnetic field model
Attractive forces can be produced on each particle of the suspension due to the interaction between the polarized nanoparticles with the external magnetic field. These external magnetic forces can be represented as body-forces on the suspension per unit volume [22], thus, is the magnetization, and l 0 [T m A À1 ] is the magnetic permeability. v is the specific volume. Equation (8) can be rewritten as, Because v ¼ 1 q , then, oq ov ¼ À 1 v 2 , the above equation can be written as, Thus, The magnetization M is a function of H , approximated by [14], The range of a 1 [A m À1 ] is of order 10 2 À 10 5 , and the order of b 1 [m A À1 ] is of order 10 À7 À 10 À3 . The two parameters a 1 and b 1 depend on the ferromagnetic material type. Moreover, they can control the saturation magnetization and the initial susceptibility. In order to simplify equation (11), let us do the followings, oM oq From equation (12), one obtains, and, oM ob 1 The density of the magnetized fluid q is assumed to be changed slightly from the non-magnetized density q 0 . The partial derivatives, oa 1 oq and ob 1 oq are approximated around the points of non-magnetized case, i.e. ða 1;0 ; b 1;0 Þ ¼ ð0; 0Þ, thus, Therefore, equation (11) can be written as, Assuming that the variation in density is very small, therefore, the expression for the magnetic force may be rewritten as, Evaluating each integral term separately, Then, sum the above two integrals to get, Therefore,

Polymer viscosity
The polymer solution viscosity l e [kg m À1 s À1 ] can be calculated by a third-order polynomial as described by the modified Flory-Huggins equation [25], where l w [kg m À1 s À1 ] is the viscosity of the aqueous phase (water), c [m 3 m À3 ] is the polymer concentration and should be determined by experiments.

Initial and boundary conditions
The initial conditions are given as, where h is the length. The boundary conditions are, such that c 0 [m 3 m À3 ] is the polymer concentration on the inlet. Thus, the specified constant concentration at the left boundary is given. When the source is located on the boundary ( dc dx ¼ 0) is somewhat misleading, because the symmetry of the Gaussian curve about its center allows ( dc dx ¼ 0) to be satisfied even as mass leaves the real domain. Analytical solutions that satisfy the no-flux boundary condition are found using the principle of superposition.

Analytical solution
The 1D x-component of F mag [N], equation (15), can be written as, or, The term oM oH represents the magnetic susceptibility which is the substance magnetic response to the magnetic field. Now, let us write the momentum equation, equation (7), in 1D form as, Integrating equation (21) over x and using the boundary condition (u ¼ u 0 at x ¼ 0), one obtains, Therefore, dp dx Then integrate once more, Now, let us calculate the first term, It is easy to find, In order to estimate the integration R x 0 l e ðcÞdx in a simple way, we assume that concentration in the flow equation is mainly driven by advection and diffusion effect is ignored thus, This is a linear advection equation has the following solution, with the initial condition c i . Also, we approximate the viscosity, Therefore, such that, Finally, one can write, But in the current case of study the initial concentration of the polymer is zero, i.e. c i ¼ 0. Therefore, Substituting the above terms in equations (26) and (33) into equation (24), one obtains, In order to determine the integration constant we use the boundary condition, p ¼ p h at x ¼ h. Therefore, Here, we calculate the pressure derivative which will be used in the velocity calculations. We used equation (12) for the definition of M ðH Þ ¼ a 1 tan À1 b 1 H , dp dx The velocity is calculated with the aid of equations (21), (29) and (35) as, The magnetic field strength in 1D may be written as follows [14], where L [m] is the distance between the poles of the magnet, and  (4), we find, Using Laplace transforms and similar to reference [26], the solution of the above equation has been obtained as, such that, Again, as the initial concentration of the polymer is zero, i.e. c i ¼ 0, we have, From the definition of the complementary error function erfc(y), one can write, e Ày 2 dy: In the case of advection absence, i.e., the above function will be reduced to the following simpler form, or,

Model validation
In order to validate solutions, they should be compared to a given known solution. However, as there are no published solutions for the current model, we can compare the reduced model of some simple cases.

Flow model validation
In the case of magnetic field absence, the pressure equation (35) reduces into the linear form, which is the solution of the original pressure equation, which is coupling of momentum equation and the original Darcy's law, i.e., du dx ¼ gðxÞ; ð49Þ One can easily integrate equation (48) to obtain the linear pressure formulation (47). Also, the viscosity is treated as a constant.

Polymer model validation
Similar to the flow model reduction in the previous subsection, we do the same thing here. Recall the concentration equation (39), for a simple solute not a polymer (R / ¼ /, u ¼û/), it may take the form [26], This is the traditional advection-diffusion equation with the well-known solution [26], If the initial concentration is zero, i.e. c i ¼ 0, we have, or if one substitutes the error functions, gets, For the pure diffusion (û ¼ 0) this formula reduces to, or, All the above concentration formula are well known solution for the concentration equations can be found in any common book such as [26].

Results and sensitivity analysis
In this section, we present a sensitivity analysis of the model parameters to study how they affect the results. In order to do this kind of analysis, the main dependent physical variables, p, u and c are examined against the physical primary parameters, a 1 ; b 1 ; B r ; k; D; f and d. Figure 2 illustrates the effects of the magnetic parameter a 1 and permeability on the velocity profiles along the core length at t ¼  Figure 3. This figure indicates that as the magnetization increases the concentration increases. The magnetization has a significant effect on concentration for a big permeability, which is diminished for small values of permeability. Also, the same figure shows the distribution of u and c against the core length, x [m]. The velocity and concentration are smaller as they get farther away from the source of velocity and concentration. In order to see the effect of a 1 on the pressure profiles we used the same above parameters except b 1 ¼ 3 Â 10 À3 . In Figure 4 the pressure profiles against the core length, x [m]. It can be seen from this figure that both permeability and magnetization reduces the pressure. In Figure 5 the effects of the diffusion coefficient D and the permeability k on the concentration profiles along the core length are presented at the above stated parameters with a 1 = 3 Â 10 2 [A m À1 ]. It is interesting to notice that for high permeabilities the advection dominates, and the opposite is true for lower permeabilities the diffusion effect takes place. In order to study the effect of the universal parameter R / on the concentration distribution, we discuss the effect of its primary parameters, namely, d; c ad;max ; f ; /; q r and B.  Figure 7

Conclusion
In this work, we developed a model to describe the transport of magnetic polymer in porous media. The analytical solution has been obtained. A sensitivity analysis of the model parameters has been presented. The magnetization has a positive effect on the polymer concentration in particular for high-permeability media, and enhances the flow. One of the interesting results is that for high permeabilities the advection dominates, and for lower permeabilities the diffusion effect takes place. The adsorption parameters are negatively affecting the concentration distribution of the polymer solution. Moreover, it was found that as the porosity increases the concentration increases, while, and as the inaccessible pore volume parameter increases the concentration decreases.