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# Open Physics

### formerly Central European Journal of Physics

Editor-in-Chief: Seidel, Sally

Managing Editor: Lesna-Szreter, Paulina

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Volume 15, Issue 1

# Numerical simulation on ferrofluid flow in fractured porous media based on discrete-fracture model

Tao Huang
/ Jun Yao
/ Zhaoqin Huang
• Corresponding author
• Department of Petroleum Engineering, Colorado School of Mines, Golden, 80401 CO., United States
• Email
• Other articles by this author:
/ Xiaolong Yin
• Department of Petroleum Engineering, Colorado School of Mines, Golden, 80401 CO., United States
• Other articles by this author:
/ Haojun Xie
/ Jianguang Zhang
Published Online: 2017-06-16 | DOI: https://doi.org/10.1515/phys-2017-0041

## Abstract

Water flooding is an efficient approach to maintain reservoir pressure and has been widely used to enhance oil recovery. However, preferential water pathways such as fractures can significantly decrease the sweep efficiency. Therefore, the utilization ratio of injected water is seriously affected. How to develop new flooding technology to further improve the oil recovery in this situation is a pressing problem. For the past few years, controllable ferrofluid has caused the extensive concern in oil industry as a new functional material. In the presence of a gradient in the magnetic field strength, a magnetic body force is produced on the ferrofluid so that the attractive magnetic forces allow the ferrofluid to be manipulated to flow in any desired direction through the control of the external magnetic field. In view of these properties, the potential application of using the ferrofluid as a new kind of displacing fluid for flooding in fractured porous media is been studied in this paper for the first time. Considering the physical process of the mobilization of ferrofluid through porous media by arrangement of strong external magnetic fields, the magnetic body force was introduced into the Darcy equation and deals with fractures based on the discrete-fracture model. The fully implicit finite volume method is used to solve mathematical model and the validity and accuracy of numerical simulation, which is demonstrated through an experiment with ferrofluid flowing in a single fractured oil-saturated sand in a 2-D horizontal cell.

At last, the water flooding and ferrofluid flooding in a complex fractured porous media have been studied. The results showed that the ferrofluid can be manipulated to flow in desired direction through control of the external magnetic field, so that using ferrofluid for flooding can raise the scope of the whole displacement. As a consequence, the oil recovery has been greatly improved in comparison to water flooding. Thus, the ferrofluid flooding is a large potential method for enhanced oil recovery in the future.

PACS: 47.11.Df; 47.54.Bd; 47.65.Cb

## 1 Introduction

Ferrofluids are stable colloids composed of small (3–15 nm) solid, magnetic, single-domain particles coated with a molecular layer of a dispersant and suspended in a liquid carrier. Thermal agitation keeps the particles suspended because of Brownian motion, and the coatings prevent the particles from sticking to each other [1]. Ferrofluid has been the subject of various experimental and numerical studies. Ghasemian et al. investigated the cooling of a water-based ferrofluid with non-coated magnetic nanoparticles in a mini channel under the influence of both constant and alternating magnetic fields [2]. Hayat et al. studied the flow of ferrofluid between two parallel rotating stretchable disks with different rotating and stretching velocities [3]. Yasmeen et al. analyzed the two dimensional ferrofluid flow with magnetic dipole and homogeneous-heterogeneous reactions [4]. Rahimi et al. measured the surface tension of a ferrofluid usingsessiledrop and falling drop method [5]. Odenbach and Thurm studied the magneto-viscous effects in ferrofluids [6].

As a new kind of functional material, ferrofluid exhibits the characteristics of a general fluid that its motion follows the hydrodynamic law. Secondly, it is a magnetic substance which receives the magnetic body force in the presence of a gradient in the magnetic field strength, so that the attractive magnetic forces allow the ferrofluid to be manipulated to flow in any desired direction through control of the external magnetic field without any direct physical contact [7]. Therefore, a ferrofluid has many industrial applications [8, 9], such as dynamic sealing, heat dissipation, inertial and viscous damper. In recent years, a controllable ferrofluid has caused the extensive concern in oil industry as a new functional material [1015], many scholars studied its potential application in enhanced oil recovery, fracture detection, etc.

As we know, water flooding is an efficient approach to maintain reservoir pressure and has been widely used to enhance oil recovery. However, the strongly heterogeneous reservoirs can significantly decrease the sweep efficiency. Therefore, the utilization ratio of injected water is seriously affected [1618]. In this paper, the potential application of using the ferrofluid as a new kind of displacing fluid for flooding in fractured porous media has been studied. The results showed, that using ferrofluid with the magnetic field for flooding can raise the scope of the whole displacement, as a consequence, the oil recovery has been greatly improved in comparison to water flooding.

## 2 Magnetic body force

Ferrofluid’s macroscopic magnetic properties come from its internal magnetic solid particles. When there is no external magnetic field present, the magnetic moments of particles (that is, the tiny magnetic field) are disordered, due to the influence of thermal motion and cancel each other; when an external magnetic field is applied, the magnetic moments are arranged neatly in the direction of the external magnetic field, so that the ferrofluid exhibits magnetism at the macroscopic level, as shown in Figure 1.

Figure 1

The components and magnetization of a ferrofluid. (a) Particle coated with a molecular layer of a dispersant; (b) Magnetization process; (c) TEM micrograph of a ferrofluid sample [19]; (d) A bottle of ferrofluid Hinano-FFW

With the increase of the external magnetic field strength H, the magnetization M of the ferrofluid increases and reaches a maximum value, that is, the saturation magnetization Mmax. In this paper, the water-based ferrofluid Hinano-FFW has been studied experimentally and theoretically by numerical simulation. The saturation magnetization Mmax = 1.596 × 104 A/m, and the magnetization curve are shown in Figure 2.

Figure 2

Magnetization versus magnetic field strength for Hinano-FFW

Magnetization curves can be approximated by simple two-parameter arctangent functions of the form [20]: $M=α×arctan⁡(β×H)$(1)

For the Hinano-FFW ferrofluid, α = 1 × 104, β = 3.5 × 10−5, besides, in the case of the immiscible two-phase flow in porous media, assuming ferrofluid magnetization increases linearly with its saturation: $M(Sff)=M(Sff=1)Sff$(2)

In this paper, the external magnetic field is provided by the NdFeB magnets. The specific parameters of the magnet are shown in Table 1, and the three-dimensional magnetic field strength H = (Hx, Hy, Hz) can be calculated by analytic equations [21]: $Hd=Gd(x,y,z)−Gd(x+L,y,z),d=x,y,z$(3) Where, $Gxx,y,z=Br4πμ0⋅arctany+az+bxy+a2+z+b2+x21/2+arctany−az−bxy−a2+z−b2+x21/2−arctany+az−bxy+a2+z−b2+x21/2−arctany−az+bxy−a2+z+b2+x21/2$(4) $Gyx,y,z=Br4πμ0⋅lnz+b+z+b2+y−a2+x21/2z−b+z−b2+y−a2+x21/2×z−b+z−b2+y+a2+x21/2z+b+z+b2+y+a2+x21/2$(5) $Gzx,y,z=Br4πμ0⋅lny+a+z−b2+y+a2+x21/2y−a+z−b2+y−a2+x21/2×y−a+z+b2+y−a2+x21/2y+a+z+b2+y+a2+x21/2$(6)

Table 1

The properties of permanent magnet used for experiment and numerical simulation

Where Br is magnet residual flux density and 2a, 2b, L, are the lengths of the magnet in three directions, respectively, as shown in Figure 3.

Figure 3

Local coordinate system for the magnetic field produced by a permanent magnet

When an external magnetic field is applied, the secondary magnetic field produced by the magnetic particles inside the ferrofluid interacts with the external magnetic field, which causes the ferromagnetic fluid to be affected by the magnetic field force. As a result, the ferrofluid is affected by the magnetic body force [22]: $Fm=μ0M⋅∇H$(7) where μ0 = 4π × 10−7 T m/A is the magnetic permeability of free space. Generally, the direction of magnetization of a ferrofluid element is always in the direction of the local magnetic field, then $Fm=μ0MHH⋅∇H=μ0MH12∇H⋅H−H×∇×H$(8)

Assuming the ferrofluid is electrically non-conducting and that the displacement current is negligible, so that ∇ × H = 0, we can obtain $Fm=μ0M∇H$(9)

## 3 Discrete-Fracture Model

Usually, fractures have complicated geometric configuration due to the various generation environment, such as stress, deposition, erosion, effloresce, etc. Thus, it is necessary to simplify the fractures for convenience. For a laminar flow conditions, velocity distribution along the fracture aperture can be obtained. Rewriting the flux in the form of equivalent Darcy’s law gives the fractures’ equivalent permeability. Evidently, the flow parameters and correlative physical quantities are kept constant along the direction of the fracture aperture, so reducing its dimension is feasible. In this paper, we use discrete-fracture model to simplify the fractures geometric configuration [23]. For the 2-D problem, Delaunay triangular mesh is employed to subdivide the whole research region and 1-D line element is employed to represent fracture. For the 3-D problem, Delaunay triangular mesh is used to subdivide the fracture surface; the entire research region is subdivided by relevant tetrahedron or hexahedron, as shown in Figure 4.

Figure 4

Mesh schematics of discrete-fracture model [23]; (a) the 2-D problem; (b) the 3-D problem

The matrix system comprising of micro-fissure and rock mass is regarded as an equivalent porous continuum and the macroscopic fractures are manifestly represented as discrete fractures. Therefore, the whole fractured porous media consist of a matrix system and fracture system. The research region is Ω = Ωm + ∑ αi × (Ωf)i, where m represents matrix, f represents fracture, and αi is the aperture of the i-th fracture. Assuming the representative element volumes of both matrix and fracture system, the flow equations (FEQ) are applicable to the entire research area. Then, for the discrete-fracture model, the integral form of the flow equation can be expressed as: $∫ΩFEQdΩ=∫ΩmFEQdΩm+∑iαi×∫ΩfiFEQdΩfi$(10)

## 4.1 Flow equations

For simplicity, we only consider isothermal flow of impressible fluid and neglect capillary pressure in this paper, which is similar to the analysis of other flow problems. From the law of mass conservation, we know that a fluid in a control volume should meet: $∫∂V−ρβvβ⋅ndA+∫VqmβdV=∫V∂∂t(ρβϕSβ)dV$(11) where ρβ is fluid density, vβ is seepage velocity, n is the outer normal unit vector of outer boundary ∂ V, qmβ is the source term which represents mass change in time unit and volume unit, ϕ is porosity of porous media, Sβ is saturation.

According to Darcy’s law, the seepage velocity is written as: $vβ=−krβμβk⋅(∇pβ−ρβg∇D)$(12) where k is permeability tensor which changes into scalar k in isotropic porous media, k is the relative permeability, μβ is fluid viscosity, pβ is fluid pressure, D denotes highness, which is positive on the upward side, g is gravitational acceleration. Particularly, for a ferrofluid affected by an external magnetic field a body force is produced, so that an additional magnetic force term appears in the Darcy’s Equation [20]: $vff=−krffμffk⋅∇pff−ρffg∇D−μ0M∇H$(13) where the subscript ffstands for ferrofluid.

## 4.2 Finite volume discretization

Using the cell center point value to represent the average value of physical quantities, Eq. (11) can be further described as: $∂∂t(ρβϕSβVi)=−∑j∈ηinij⋅(ρβvβ)Aij+qmβVi$(14) where j is the adjacent cell of i, nij is the outer normal unit vector of interface between i and j, Aij is the interface area. Substituting Eq. (12) for (14), we can obtain: $∂∂t(ρβϕSβVi)=∑j∈ηinij⋅ρβkrβμβk⋅∇ΦβAij+qmβVi$(15) Herein, we define the flow potential as follows: $Φβ=pβ−ρβgD$(16) In particular, from Eq. (13) we know the ferrofluid flow potential can be written as: $Φff=pff−ρffgD−μ0MH$(17)

Using the first order difference for time discretization and central difference for space discretization, the Eq. (15) can be further written as: $1Δt(ρβSβϕV)in+1−(ρβSβϕV)in=∑j∈ηi(ρβλβ)ij+1/2n+1γijΦβjn+1−Φβin+1+Qβin+1$(18)

where $\begin{array}{}\left({\rho }_{\beta }{\right)}_{ij+1/2}^{n+1}=\left[\left({\rho }_{\beta }{\right)}_{i}^{n+1}+\left({\rho }_{\beta }{\right)}_{j}^{n+1}\right]/2\end{array}$ is the average density at interface between i and j, γij = Aijkij+1/2/(di+dj) is the conductivity, kij+1/2 is the harmonic average of permeability ki and kj, di and dj is the vertical distance from interface to the center point of i and j, respectively. Source term $\begin{array}{}{Q}_{\beta i}^{n+1}=\left({q}_{\text{m}}\beta {\right)}_{i}^{n+1}={\rho }_{\beta }{q}_{\beta i}^{n+1}\phantom{\rule{thinmathspace}{0ex}}{V}_{i},\end{array}$ λβ = k/μβ is the mobility coefficient and the upstream calculation formula is implemented as follows: $(ρβλβ)ij+1/2n+1=(ρβλβ)in+1if(Φβjn+1−Φβin+1)≤0(ρβλβ)jn+1if(Φβjn+1−Φβin+1)>0$(19)

## 4.3 Solving the discrete equation

The Newton-Raphson iterative method is used to solve the numerical discrete equation in this paper. The numerical discretized mass conservation Equation (18) is written in the following residual form: $Rβin+1=1Δt(ρβSβϕV)in+1−(ρβSβϕV)in−∑j∈ηi(ρβλβ)ij+1/2n+1γij(Φβjn+1−Φβin+1)−ρβqβin+1Vi$(20)

Expanding to the first order in the primary variables and introducing an iteration index p, we can obtain $∑l∂Rβin+1(xp)∂xlpδxlp+1=−Rβin+1(xp)$(21) where $\begin{array}{}{x}_{l}^{p}\end{array}$ is the l-th primary variable at p-th iteration level, time step $\begin{array}{}n+1,\delta {x}_{l}^{p+1}={x}_{l}^{p+1}-{x}_{l}^{p}\end{array}$ is the increment of primary variable at iteration level p + 1. Solution of the Jacobian matrix system of equations is obtained with the last iteration level, resulting in an updated estimation of the primary variables. Iteration continues until the latest residuals are reduced to a small value, $|Rβip+1,n+1|<ε$(22) In this paper, convergence criterion ε = 1 × 10−5.

## 5 Examples and discussions

(1) Single fractured porous media model

Considering the single fractured porous media model in Figure 5a, the porosity of the homogeneous isotropic matrix ϕ = 0.2, permeability km = 1.38 × 10−12 m2, fracture aperture α = 1 mm, and permeability kf = α2/12 = 8.33 × 10−8 m2. Viscosity of ferrofluid μff = 5.8 mPa•s, viscosity of oil μo = 22.1 mPa⋅s. Density of ferrofluid ρff = 1187 kg/m3, density of oil ρo = 850 kg/m3, ferrofluid phase relative permeability $\begin{array}{}{k}_{r\text{ff}}={S}_{\text{ff}}^{2},\end{array}$ oil phase relative permeability kro = (1 − Sff)2; we assumed irreducible ferrofluid saturation and residual oil saturation are equal zero. Initial oil saturation is equal 1, both injection and producing speeds are q = 0.01Vp/min, where Vp is the total pore volume. The triangular meshes consist of 603 nodes and 1124 elements as shown in Figure 5b. Before the injection of the ferrofluid, a magnet PM1 was put on the upper and the right side of the model and the magnetic field, as shown in Figure 5c.

Figure 5

Single fractured porous media model; (a) Model geometry; (b) Triangular meshes; (c) Distribution of magnetic field H in units of Gs

The ferrofluid saturation distribution of flow experiment and numerical simulation on the single fractured model are shown in Figures 6 and 7. There are some differences between calculated and experimental results, because it is impossible to make extreme homogeneous isotropic matrix. However, it still can be seen that the calculation result is basically consistent with the experimental result, which verifies the validity and accuracy of the mathematical model and numerical algorithm.

Figure 6

The ferrofluid saturation distribution of flow experiment and numerical simulation at different injection volume; (a)-(a”) respectively show the experimental results at 0.25Vp, 0.75 Vp, 1.5Vp; (b)-(b”) respectively show the calculation results at 0.25Vp, 0.75Vp, 1.5Vp

Figure 7

The ferrofluid-flooding production index curves of experiment and simulation; (a) Ferrofluid cut curves; (b) Recovery curves

(2) Complex fractured porous media model

As Figure 6 shows, affected by the attractive magnetic forces, the ferrofluid was manipulated to flow to the magnets. Thus, we designed a complex fractured porous media model which has multiple fractures in the lower part of the model, and a magnet PM2 was put on the left-top of the model, as shown in Figure 8. Next, we simulated water flooding and ferrofluid flooding process on this model, to study the potential by using the ferrofluid as a displacing fluid for flooding in fractured porous media. The viscosity of water μw = 1 mPa⋅s and ρw = 1000 kg/m3, both injection and producing speeds are q= 0.01Vp/min.

Figure 8

Complex fractured porous media model; (a) Model geometry; (b) Distribution of magnetic field H in units of Gs

As shown in Figure 9, the most amount of injected water flows into fractures during the water flooding process, because the fractures provide high-conductivity paths. As a result, a portion of the remaining oil has not been displaced (especially in the left-upper part of model) and the flooding sweep area has become smaller. However, the ferrofluid was controlled to flow into the low sweep area, when the magnetic field was applied during the ferrofluid flooding process, leading to the displacement of most amount of the oil.

Figure 9

The saturation distribution at different injection volume during water flooding and ferrofluid flooding process; (a)-(a”) respectively show the water saturation distribution at 0.25 VP, 0.75 VP, 1.5 VP; (b)-(b”) respectively show the ferrofluid saturation distributionat 0.25 VP, 0.75 VP, 1.5 VP

As seen above, using ferrofluid with external magnetic field for flooding can expand the sweep area and enhance the displacement efficiency. Thus, the recovery ratio improved from 40% to 62% compared to water-flooding, as shown in Figure 10.

Figure 10

The production index curves of water-flooding and ferrofluid-flooding; (a) Water cut and ferrofluid cut curves; (b) Recovery curves of water-flooding and ferrofluid-flooding

## 6 Conclusions

1. In this paper, the potential application of using the ferrofluid as a new kind of displacing fluid for flooding in fractured porous media has been studied for the first time. Using the fully implicit finite volume method to solve mathematical model, the validity and accuracy of numerical simulation is demonstrated through an experiment, in which ferrofluid flows in a single fractured oil-saturated sand in a 2-D horizontal cell. At the end, the displacement effect between water flooding and ferrofluid flooding on a complex fractured porous media has been studied.

2. The calculation results showed the magnetic force can control ferrofluid flow in desired direction. Therefore, when there is a high conductivity path, such as high permeability zone or fracture, using ferrofluid for flooding, one can raise the scope of the whole displacement. As a consequence, the oil recovery has been greatly improved compared to water flooding. Thus, the ferrofluid flooding is potentially a promising method for enhanced oil recovery in the future.

3. In this paper, only 2-D problem was discussed. 3-D problem and multiphase problem are the next research projects.

## Acknowledgement

The authors would like to express their gratitude to the National Natural Science Foundation of China (No. 51234007, 51490654, 51404292, 41502131) and The Fundamental Research Funds for the Central Universities (No. 16CX06026A) for their support.

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Accepted: 2017-01-20

Published Online: 2017-06-16

Citation Information: Open Physics, Volume 15, Issue 1, Pages 370–378, ISSN (Online) 2391-5471,

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