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

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 [10–15], 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 [16–18]. 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 M_max. In this paper, the water-based ferrofluid Hinano-FFW has been studied experimentally and theoretically by numerical simulation. The saturation magnetization M_max = 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]:

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:

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 = (H_x, H_y, H_z) can be calculated by analytic equations [21]:

Where,

Where B_r 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]:

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

Assuming the ferrofluid is electrically non-conducting and that the displacement current is negligible, so that ∇ × H = 0, we can obtain

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 = Ω_m + ∑ α_i × (Ω_f)_i, where m represents matrix, f represents fracture, and α_i 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:

4 Numerical simulation of ferrofluid flow in porous media

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 k_m = 1.38 × 10⁻¹² m², fracture aperture α = 1 mm, and permeability k_f = α²/12 = 8.33 × 10⁻⁸ m². Viscosity of ferrofluid μ_ff = 5.8 mPa•s, viscosity of oil μ_o = 22.1 mPa⋅s. Density of ferrofluid ρ_ff = 1187 kg/m³, density of oil ρ_o = 850 kg/m³, ferrofluid phase relative permeability krff=Sff2, oil phase relative permeability k_ro = (1 − S_ff)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 = 0.01V_p/min, where V_p 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 0.25V_p, 0.75 V_p, 1.5V_p; (b)-(b”) respectively show the calculation results at 0.25V_p, 0.75V_p, 1.5V_p

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 μ_w = 1 mPa⋅s and ρ_w = 1000 kg/m³, both injection and producing speeds are q= 0.01V_p/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 V_P, 0.75 V_P, 1.5 V_P; (b)-(b”) respectively show the ferrofluid saturation distributionat 0.25 V_P, 0.75 V_P, 1.5 V_P

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.