1) The fluid system includes the liquid phase and gas phase (foams), and the non-Newtonian fluid is characterized by a pressure gradient function. 2) Liquid phase of enhanced foam system, as incompressible power-law fluid, is polymer and surfactant solution. The surfactant concentration is constant and the gas phase is regarded as an ideal gas. 3) Surfactant is only found in water phase. Due to its small concentration, the effects on the density and viscosity of each phase can be ignored. 4) Using a modified gas phase relative permeability model to characterize the relative permeability of the foam.

1) Gas-liquid two phase mass conservation equation

Gas-liquid two phase mass conservation equation of the three dimensional unsteady flow is an equation about the enhanced foam system in the porous media.

The gas phase:

(1)

The liquid phase:

(2)

Where ϕ is the core porosity, dimensionless; ρ_g is the gas phase density, kg/m³; µ_g is the Darcy rate of gas phase, m/s; q_g is the gas injection rate, kg/m³s; S_g is the gas phase saturation, dimensionless; ρ_p is the liquid density, kg/m³; µ_p is the Darcy rate of liquid phase, m/s; q_p is liquid injection rate, kg/m³.s; S_p is liquid saturation, dimensionless.

2) Gas-liquid two phase motion equations

The gas phase:

(3)

The liquid phase:

(4)

Where K is the core permeability, m²; K_rg is the gas phase relative permeability, dimensionless; P_g is the liquid pressure, Pa; μ_g is the gas phase viscosity, Pa.s; K_rp is the liquid phase relative permeability, dimensionless; P_p is the liquid pressure, Pa; μ_p is the liquid apparent viscosity, Pa.s, D is the vertical depth of oil reservoir.

(1) The gas phase relative permeability:

(5)

Where is the calculation gas phase relative permeability coefficient, dimensionless; n_g is the calculation gas phase relative permeability power exponent, dimensionless, K _rf is the foam effective permeability, dimensionless.

(2) The liquid phase relative permeability:

(6)

Where is the calculation liquid phase relative permeability coefficient, dimensionless; S_pc is the bound liquid phase saturation, dimensionless; n_l is the calculation liquid phase relative permeability power exponent, dimensionless.

(3) Liquid apparent viscosity

Apparent viscosity expression of power law fluid:

(7)

Where n is the power law fluid liquidity index, dimensionless; K_c is the power law fluid consistency coefficient, Pa.s"; c' is the pore tortuosity coefficient, dimensionless.

3) State equation

The gas phase state equation is:

(8)

(9)

Where P_ga is the pressure under the condition of the standard atmospheric pressure, Pa; V_a is the gas volume under the condition of the standard atmospheric pressure, m³; P_g is the gas phase pressure, Pa; V is the gas volume under the gas phase pressure, m³; ρ_ga is the gas density under the condition of the standard atmospheric pressure, kg / m³.

4) The basic differential equation

Take the motion equations (3) and (4) into the mass conservation equations (1) and (2), and then the basic differential equation of gas-liquid two phase of the three-dimensional unsteady flow about the enhanced foam system in the porous media is given.

The gas phase:

(10)

The liquid phase:

(11)

Only using the gas liquid two phase basic differential equation to describe the flow law of enhancing foam system in porous media is not enough, the structure change of foam when it flows in porous media must be considered, such as the production, burst, coalescence, migration and retention, start of foam and so on. The foam amount balance equation describes the number and size of foam change and its effect on gas phase relative permeability and the effective viscosity through the average density of flowing foam and the average density of interception foam.

Based on the previous; research results, the enhanced foam system amount balance equation under the condition of the average foam size is [9Kovscek, A.R.; Radke, C.J. Fundamentals of Foam Transport in Porous Media. Foams: Fundamentals and Applications in the Petroleum Industry; American Chemical Society: Washington, DC, 1994, pp. 115-163.
[http://dx.doi.org/10.1021/ba-1994-0242.ch003] -16Teng, L.; Zhaomin, L.; Jing, L.; Ran, L. A mathematical model of foam flooding based on foam microscopic seepage characteristics. Chinese. J. Comput. Phys., 2012, 29(04), 519-524.]:

(12)

Where n_f, n_t is the average density of the flowing foam and the trapping foam; X_f, X_t is the shunt volume of the flowing foam and the trapping foam, u_f is the Darcy flow rate of gas; k₁ is the foam formation rate constant; k_-1 is the coalescence rate constant; v_w, v_f is the real flow rate of liquid and gas; is the gas injection(output) rate; is the flowing foam rate in well point.

Coalescence rate constant is:

(13)

P_c is the capillary force. Expression of critical capillary force is:

(14)

Where C_s, C_s⁰ is the surfactant concentration and concentration reference value of liquid phase; P_c^*, max is maximum critical capillary force.

Calculation formula of foam viscosity as follows [17Khatlb, Z.I.; Hirasaki, G.J.; Falls, A.H. Effects of capillary pressure on coalescence and phase mobilities in foams flowing through porous media. Soc. Petrol. Eng., 1988, 919-926.]:

(15)

Where α is the viscosity coefficient which is related to the structure and concentration of surfactant.

The fraction of trapping foam in the foam can be written as:

(16)

Where X_t,max is the maximum trapping foam fraction, β is the empirical constant.

1) The equilibrium relation between the flowing foam and the tarpping foam:

(17)

2) The saturation relationship:

(18)

3) Capillary pressure equation:

(19)

Using IMPES(implicit pressure explicit saturation) method to calculate the mathematical model of three dimensional unsteady flow of enhanced foam system in porous medium. This method is implicit pressure explicit saturation method. That is, it can use implicit expression to solve pressure equation, and show the equation of solving saturation.

In order to verify the correctness of the model, firstly, the experiment of enhanced foam flooding after water flooding was carried out. The experimental system and flow chart of N₂ foam flooding is shown in Figs. (1 and 2).

Fig. (1) The experimental system of foam flooding.

1 Injection pump 2 Flowmeter 3 N₂ 4 Valve 5 Foamer 6 Oil 7 Formation water 8 Pressure gauge

9 Core 10 Back pressure valve 11 Separator

Fig. (2) The flow chart of N2 foam displacement experiment.

Experimental methods: the process of core flooding was connected, installed and debugged, and the artificial quartz sand model (size 25mm×300mm, permeability about 1.5μm², pore volume about 55cm³) was inserted into the process, then the water was injected at speed of 0.5 ml/min until the flooding pressure steadied, then the concentration 0.5% foam agent solution mixed with nitrogen was injected into the core by six-way valve, and the pressure difference at both ends of the core was recorded until it stable, and the experimental data are shown in Fig. (3).

The experimental results of enhanced foam flooding after water flooding were calculated and fitted. The foam is injected with the polymer solution which concentration is 1250mg/L. As a result of the existence of polymer, the stability of the enhanced foam system is enhanced which can make the foam exist steadily in oil model. In enhanced foam flooding experiment, the enhanced foam system is injected after water flooding, the differential pressure rising rapidly. Fitting results are shown in Fig. (3). It can be seen from the fitted curve that mathematical model calculation value and experimental value are consistent.

Fig. (3) Fitting curve of enhanced foam flooding about oil pressure differential.

The distribution of average density of the flowing foam, which is unsteady flow of enhanced foam system in porous media, is shown in Fig. (4). It can be seen from Fig. (4) that the unsteady flow of the enhanced foam system in porous media exists flow front. The average density of flowing foam of the flow front reaches the peak. When the liquid saturation of flow front steep rises to 1.0, the capillary force falls, the structure of foam becomes thinner and it leads to the average density of flowing foam appeared peak. In addition, the average density peak of the flowing foam of the flow front increases with the injection time. This is because the foam piston strength increases with the injection time.

Fig. (4) The distribution of the average density of the flowing foam.

Fig. (5) is the core phase saturation distribution after enhanced foam flooding 0.2 PV. It can be seen that due to the high viscosity characteristics of the foam, the steady foam system is formed at the injection side, the oil saturation drop to around 0.1 in the part where is displaced. In the displacement front, the oil bank is formed. The oil saturation of oil bank reaches about 0.55. The oil saturation which is residual oil saturation after water flooding in outlet is about 0.38.

Fig. (5) Oil phase saturation distribution of enhanced foam flooding.