#### Abstract

Based on the features of microemulsion flooding in low-permeability reservoir, a three-dimension three-phase five-component mathematical model for microemulsion flooding is established in which the diffusion and adsorption characteristics of surfactant molecules are considered. The non-Darcy flow equation is used to describe the microemulsion flooding seepage law in which the changes of threshold pressure gradient can be taken into account, and the correlation coefficients in the non-Darcy flow equation are determined through the laboratory experiments. A new treatment for the changes of threshold pressure and the quantitative description of adsorption quantity of surfactant and relative permeability curves are presented, which enhance the coincidence between mathematical model and experiment results. The relative errors of main development indexes are within 4%. A software is programmed based on the model to execute a core-level small-scale numerical simulation in Chaoyanggou Oilfield. The fitting relative errors of the pressure, flow rate, and moisture content are 3.25%, 2.71%, and 2.54%, respectively. The results of laboratory experiments and numerical simulation showed that microemulsion system could reduce the threshold pressure gradient by 0.010 MPa/m and injection pressure by 0.6 MPa. The biggest decline in moisture content reaches 33%, and the oil recovery is enhanced by 10.8%.

#### 1. Introduction

With the characteristics of narrow pores, large pore-throat ratio, high irreducible water saturation, and serious heterogeneity, the low-permeability reservoirs do not meet Darcy’s law anymore and there is threshold pressure gradient. Compared to the high-pressure of water injection, low water displacement efficiency, and low ultimate recovery efficiency of water flooding [1–4], the surfactant flooding can reduce the threshold pressure gradient and injection pressure, increase the water injection rate which expands the swept volume, reduce the interfacial tension, and improve the displacement efficiency [5–8]. According to the surfactant concentration in the injection system, the surfactant flooding is divided into active water system (*C*_{s} < 1%) and microemulsion system (*C*_{s} = 3%∼5%) [9–13]. Because of the peculiarities of ultra-low interfacial tension and better mobility control, the latter one has become the hotspot in enhanced oil recovery of low-permeability reservoir. As the microemulsion flooding is characterized with high risk and high input, the numerical simulation of this method should be put to the strategic position. The leader in this aspect is the University of Texas, BYUYANHE and POPE, who established UTCHEM model [14, 15]. The biggest drawback of the models is that the model is unable to consider the threshold pressure gradient. In addition, so many factors are considered that it is difficult to get the results, and the calculation accuracy or convergence of the model is poor [16–18]. Han introduced the chemical model to the EOS composition simulator module, and a fully implicit parallel component chemical flood simulator is established to improve the calculation speed [19]. However, since some parameters are difficult to obtain, the applicability of the model is limited. In this paper, based on the characteristics of the microemulsion flooding, a three-dimension three-phase five-component microemulsion flooding mathematical model is established, in which the diffusion and adsorption of surfactant molecules, the change of microemulsion viscosity, and relative permeability are considered. The finite difference method is used to solve the model, and simulation accuracy and calculation speed are improved.

#### 2. The Non-Darcy Equation in Low-Permeability Reservoir

According to the literature [20], non-Darcy flow equation in low-permeability reservoir can be expressed as

The laboratory experiments are carried out to get the value of and . The microemulsion system used in the experiment is made of water, SDS, light oil, *n*-butanol, and sodium chloride. The experimental schemes and the values of and under different conditions are shown in Table 1, and the seepage velocity-pressure gradient curves are shown in Figure 1.

#### 3. The Establishment of Microemulsion Flooding Mathematical Model

##### 3.1. The Assumptions of the Model

Generally, oil, water, surfactant, alcohol, and inorganic salt are taken into account in the microemulsion flooding. And it is assumed that there is a three-phase flow involving oil, water, and microemulsion in the reservoir. It is considered that the reservoir is anisotropic and heterogeneous, and the reservoir rock and fluids involved in the microemulsion flooding process are compressible. The influences of capillary force and gravity effect are also included.

##### 3.2. The Material Balance Equation

Considering the diffusion of surfactant molecules among each phase and the adsorption on the rock surface, the continuity equation of the *j* composition in the *i* phase is

In above equations, indicates oil, water, and microemulsion phases, denoted by *o*, , and *m*, and the value of could be 1, 2, or 3; *j*, assigning value of 1, 2, 3, 4, or 5, represents the compositions of oil, water, surfactant, alcohol, and inorganic salt, respectively.

It is assumed that the mass percentage of water composition in oil phase, *C*_{o2} is equal to 0, and the mass percentage of oil composition in water phase, *C*_{w1} is 0, too. The *D*_{ij} and *a*_{ij}, pertaining to oil and water compositions, respectively, are neglected. Combining equations (1) and (2), a three-dimensional, three-phase, and five-composition mathematical model for microemulsion flooding can be obtained:where is the diffusion velocity of composition in phase; is the mass percentage of composition in phase; and is the comprehensive mass percentage of phase including the adsorbed terms.

##### 3.3. The Auxiliary Equation

The relationships of saturation among each phase and mass percentage of *j* composition in each phase can be written as

The relationship of pressure among each phase is as follows:where *a* = 1, 2, and 3 and when *a* = *i*.

#### 4. The Treatment of Important Parameters

##### 4.1. Interfacial Tension

Surfactant plays an important role in reducing interfacial tension and improving the flow ability of crude oil in the microemulsion system. The indoor experiments are carried out to get the interfacial tension in different surfactant concentrations. And, the results are shown in Figure 2.

It can be seen from Figure 2 that with the increase of surfactant concentration, the interfacial tension decreases greatly. But, the extent of decrease becomes smaller when the surfactant concentration reaches a certain value. The reason is that with the increase of surfactant concentration, the adsorption quantity of surfactant molecules in oil layers also increases, but the increase of effective concentration is not proportional. Therefore, the relationship between surfactant concentration and the interfacial tension could be written aswhere the value of coefficients , index , constant , and could be given through the experiments.

##### 4.2. Adsorption

The Langmuir adsorption curve is used to describe the surfactant molecules adsorption in the process of microemulsion flooding, in which the effects of salt concentration, surfactant concentration, and core permeability are considered. And the expression is

The concentration in the adsorption process is normalized by water, and the minimum value is got to make sure that the total amount of adsorption is not greater than the surfactant concentration. Adsorption quantity increased with the increase of salt concentration, but decreased with the increase of permeability:where is effective salinity, and . Adsorption parameters and can be obtained by fitting the adsorption data of surfactant measured indoor.

##### 4.3. Viscosity

The viscosity of microemulsion system is related to the components and the viscosity of oil and water. In the process of microemulsion flooding, the surfactant molecules could adsorb onto the interface in the porous medium. And some crude oil would be dissolved into the displacement phase under the action of surfactant molecules which changed the microemulsion viscosity. To research the change of microemulsion system viscosity, 1.2 mPa·s saline and 4.2 mPa·s light oil are used to formulate microemulsion system according to different oil-water volume ratio. The curve between microemulsion viscosity and oil-water volume ratio is shown in Figure 3.

The graph in Figure 3 can be described by the following formula:where is the oil-water viscosity ratio, is the oil-water volume ratio, and is the index, which is related to the character and concentration of surfactant. The value of these parameters could be obtained by experiments.

##### 4.4. Relative Permeability

Steady-state method is applied to determinate the relative permeability curves between oil-water and oil-microemulsion in different surfactant concentration. And the porous media are natural cores in Chaoyanggou Oilfield. The relative permeability curves under different surfactant concentration are shown in Figure 4.

**(a)**

**(b)**

Multiphase flow relative permeability could be simulated by the Corey function. And the impact of crude oil entrapment on the relative permeability is taken into account. It can be written aswhere denotes relative permeability endpoints for the phase, which can be obtained through the experiment.where variables with superscripts High and Low expressed the values at high and low captured number.

##### 4.5. Phase Equilibrium Constraints

Regard the fluid in the reservoir as a ternary system including water, oil, and microemulsion. And the equilibrium conditions can be described by the equilibrium theory of regular solution for ternary system. According to the thermodynamic relations, the spinodal equation for three phases can be written as

Under a certain temperature and pressure condition, the chemical potential of each component comply with the Gibbs–Duhem equation:

For the three-component system, the formulas above can be written as

Under the condition of a certain pressure,

Moreover, should satisfy the following equation:

Apply the formula above to calculate the content of oil, water, and microemulsion and obtain the equilibrium relationship:

#### 5. The Solution of Model

##### 5.1. The Solution Steps of the Mathematical Model

The finite difference method is used to solve the mathematical model. The solution steps are as follows:(1)Combining equations (3), (4), and (6) for water phase, oil phase, and microemulsion phase, get the pressure *p*_{o}, *p*_{w}, *p*_{m};(2)Substituting pressure *p*_{o}, *p*_{w}, *p*_{m}, into equation (3) for water phase to calculate water saturation *s*_{w};(3)Combining equation (3) for microemulsion phase, (5) for water phase and oil phase, and (18) to eliminate relative variables and obtain differential equation including variable *C*_{w3}, then, substituting pressure value *p*_{o} and saturation *s*_{w} into this differential equation to calculate concentration *C*_{w3};(4)Applying the same method to get *s*_{o}, *s*_{m} and *C*_{o3,}*C*_{m3}.

##### 5.2. The Implementation Process of the Mathematical Model

The flow chart of implementation process of the Mathematical Model is shown in Figure 5.

#### 6. Calculation of an Example

The indoor experiments in tablet cores and the corresponding core-level small-scale numerical simulation are carried out to study the microemulsion flooding mechanism. Block-centered grid system is established on the model for Cartesian coordinate grid, and *X* and *Y* directions are divided into 21 grids, and the grid step length is 1.5 cm. There are 1 production well and 4 injection wells in the table core, and the distance from each injection well to the production well is 21.2 cm. Mesh dissection in the model is shown in Figure 6, and the three-dimensional geological model is shown in Figure 7. The injection system is water flooding + microemulsion flooding + subsequent water flooding.

Through numerical simulation, the injection pressure curve and moisture content and oil recovery curves are obtained, as shown in Figures 8 and 9. As you can see from the figures, with the increase of water injection volume, the injection pressure continues to rise and reaches the maximum value at 1.37 MPa; after that, the pressure gradually reduces to a steady state and keeps at 1.2 MPa. The theoretical value of oil recovery is 50.517%, the operative value is 51.183%, and the relative fitting error is 1.318%. Within this stage, the water cut rises greatly, but slowly as the oil recovery degree. The reason is that the injected water rushes along the high permeability layer due to the serious heterogeneity and larger thickness of the high-permeability layer in the core. In the process of microemulsion flooding, since the surfactant molecules could reduce oil-water interfacial tension and change the wettability, the injection pressure greatly reduces to 0.37 MPa with the theoretical value of 0.40 MPa. In this period, the oil recovery increases by 12.35% while the increase of water cut is within 10%. Some crude oil is dissolved into the microemulsion and the particle size increased, which increased the microemulsion apparent viscosity and the injection pressure. The larger microemulsion particles can plug higher permeability layer effectively, and the moisture content reduced sharply. Meanwhile, with the high permeability layer plugged, the seepage resistance increased, and the following injected microemulsion would enter the low-permeable layer. That could decrease the threshold pressure gradient and injection pressure in low-permeability layer, enlarge sweep volume of displacement fluid, and increase recovery efficiency effectively. In the subsequent water flooding, injected water dilutes the microemulsion system which leads to the decrease of its viscosity and the injection pressure. With the increase of injection volume, the viscous effect of microemulsion system is not obvious. Due to the adsorption of surfactant molecules on the surface in core porosity and the change of wettability, the injection pressure of water reduces and keeps at about 0.7 MPa. Analysis of the experimental data and the simulative data through the entire process, the fitting relative errors of the pressure, flow rate and moisture content are 3.25%, 2.71%, and 2.54%, respectively, the biggest decline in moisture content is up to 33%, and the oil recovery is enhanced by 10.8%.

#### 7. Conclusion

(1)A three-dimension three-phase five-component mathematical model is established for microemulsion flooding in which the adsorption and diffusion of surfactant molecules are considered. And the finite difference method is used to solve the mathematical model.(2)Quantitative description for threshold pressure gradient, viscosity of the microemulsion, and relative permeability are given and they are used in the model. A numerical simulation software for microemulsion flooding is developed.(3)Experimental and theoretical calculations showed that microemulsion system could reduce threshold pressure gradient 0.010 MPa/m and injection pressure 0.6 MPa.(4)The biggest reduction of moisture content is 33 percentage points, and the oil recovery is improved by 10.8 percentage points; the relative errors of main development indexes are within 4%.

#### Data Availability

The data used to support the findings of this study were supplied by the corresponding author under license and so cannot be made freely available. Requests for access to these data should be made to the corresponding author.

#### Conflicts of Interest

The authors declare that they have no conflicts of interest.

#### Acknowledgments

This work is supported by the Natural Science Foundation of China under grant no. 51474071 and supported by Northeast Petroleum University Innovation Foundation For postgraduate no. YJSCX2017-006NEPU.