Foam flooding is an effective way to control water channeling and improve oil recovery in fractured low permeability reservoirs. The stability and effective time of foam are important factors affecting the effect of foam flooding^[1,2]. Improving the stability of foam in deep reservoir is the key to improving the effect of foam flooding^[3]. At present, the foam stability is mainly studied through experiments and mechanisms of reducing surface tension, changing the interfacial layer structure, enhancing the expansion of viscoelasticity by using the synergy between different chemical agents, synthesizing new foaming agents, and adding foam stabilizer etc.^[4,5,6,7]. The Marangoni effect is the phenomenon of the surfactant molecules flowing from the high concentration area to the low concentration area driven by the surface tension gradient caused by the uneven distribution of surfactants on the liquid film. It is one of the intrinsic mechanisms affecting foam stability. Schwartz^[8] and Mysels^[9] established the drainage and evolution models of vertical liquid membrane and multi-scale liquid membrane. The simplified model of the foam drainage process represented by the vertical liquid film drainage model with both ends fixed can simulate the whole process of vertical liquid membrane drainage well, and explained the Marangoni phenomenon. With the research going deeper, Ye et al.^[10,11] and Yang et al.^[12] studied the influences of active agent concentration, separation pressure and surface viscosity on the vertical liquid membrane drainage process on the basis of vertical liquid membrane drainage model, and revealed the Marangoni effect, evolution law and control factors of foam decay in the process of foam drainage.

Low frequency artificial seismic wave composite foam flooding technology is a new dynamic physical intervention method to improve the stability of foam in deep reservoir by using low frequency seismic wave. The composite technology has been tested in a single well group in Ordos Basin and achieved remarkable results^[13,14]. The subsequent research also shows that low frequency vibration can indeed improve the half-life of foam in different porous media and reduce the seepage velocity of Plateau boundary foam^[15]. However, the evolution the micro-dynamic mechanism of stability and action characteristics of foam drainage under the excitation of low-frequency artificial seismic waves need to be further studied. The low-frequency artificial seismic wave disturbance force is introduced into the vertical liquid film drainage model in this study, and the evolution model of foam vertical liquid film drainage under the excitation of low-frequency wave is established to explore the strengthening effect of low-frequency wave on the Marangoni effect during and the effect of reducing the velocity of liquid film drainage on the vertical liquid film drainage process^[16].

The foam liquid film is formed by metal wire frame and the side view is shown in Fig. 1. In order to simplify the model, the following assumptions are proposed: (1) The foam is vertical foam liquid film. (2) The surfactant solution is an incompressible fluid, with constant density and viscosity. (3) The evaporation of surfactant solution during liquid film drainage is ignored. (4) The top and bottom of the foam liquid film are fixed on the metal wire frame to meet the boundary condition of the foam liquid film without slip. (5) The thickness of the foam liquid film is symmetrical about y=0 and satisfies $y=\pm h(x, t)$, and we consider, henceforth, only the half-film $y \geq 0$. (6) The ratio of the initial thickness (h₀) of the foam liquid film to the characteristic length (L) of the foam liquid film is very small, that is, $\varepsilon=h_{0} / L$ is far less than 1, which ensures that it is suitable for lubrication theory^[17]. (7) The wave force is the sinusoidal force, and the initial direction of the wave force is assumed opposite to the direction of gravity. The expression of the wave force introduced in the paper is $\rho a \mathrm{e}^{-\alpha \kappa t} \sin (2 \pi f t)$.

According to the mass conservation equation and momentum conservation equation, the force generated by low-frequency wave is introduced into the motion equation, and the continuity equation and motion equation of fluid flow in the liquid film excited by low-frequency wave are obtained as follows:

The boundary conditions are as follows:

When y = 0

When y = h

When y = h, the tangential stress and surface pressure on the liquid film are respectively:

In the direction of x, the velocity of liquid drainage can be divided into slip velocity and shear velocity^[8]:

In Eq. (7), the surface pressure of liquid film is composed of capillary force and separation pressure:

The calculation model of separation pressure adopts the two-item separation pressure model^[11], which can be expressed as follows:

The relationship between the surface tension and surfactant concentration of the surfactant solution is as follows:

On the surface of the foam liquid film, the shear stress is approximately equal to the surface tension gradient of the foam liquid film.

Because of the Marangoni effect, the influence of surfactant concentration on slip velocity should be considered when determining the slip velocity. The convection- diffusion evolution equation of surfactant is as follows:

Define a set of characteristic terms:

$p_{\mathrm{f}}=\frac{\mu U_{0}}{L} \quad T=\frac{3 \mu L^{4}}{\sigma_{0} h_{0}^{3}}$

$U_{0}=\frac{\sigma_{0} h_{0}^{3}}{3 \mu L^{3}} \quad U_{1}=\varepsilon^{2} U_{0} \quad V=\varepsilon U_{0}$

Non-dimensionalize the above variables:

$A_{\mathrm{D}}=\frac{A}{\varepsilon^{2} \sigma_{0} / h_{0}} \quad C_{\mathrm{SD}}=\frac{K C_{\mathrm{s}}}{\sigma_{0} \varepsilon^{2}} \quad D_{\mathrm{D}}=\frac{D T}{L^{2}}$

$f_{\mathrm{D}}=T f \quad h_{\mathrm{D}}=\frac{h}{\varepsilon L} \quad p_{\mathrm{D}}=\frac{p}{p_{\mathrm{f}}} \quad p_{\mathrm{SD}}=\frac{p_{\mathrm{S}}}{p_{\mathrm{f}}}$

$p_{\mathrm{seD}}=A_{\mathrm{D}}\left[\left(\frac{h_{\mathrm{eq}}}{h}\right)^{4}-\left(\frac{h_{\mathrm{eq}}}{h}\right)^{3}\right] \quad t_{\mathrm{D}}=\frac{t}{T}$

$u_{0 \mathrm{D}}=\frac{u_{0}}{U_{0}} \quad u_{1 \mathrm{D}}=\frac{u_{1}}{U_{1}} \quad V_{\mathrm{D}}=\frac{V}{V}$

$X_{\mathrm{D}}=\frac{X}{L} \quad y_{\mathrm{D}}=\frac{y}{\varepsilon L} \quad \tau_{\mathrm{D}}=\frac{\tau}{\varepsilon \mu \frac{U_{0}}{L}}$

The dimensionless shear stress equation is obtained by substituting the characteristic term and dimensionless variables into Eqs. (1)-(8).

Based on the lubrication theory and the integral expression of mass conservation, the dimensionless liquid film thickness equation can be obtained as follows:

The dimensionless flow equation is as follows:

Eq. (11) and Eq. (12) are non-dimensionalized first, and then combined with Eq. (14) into a dimensionless equation system:

Eq. (9) and Eq. (10) are non-dimensionalized first, and then combined with Eqs. (15)-(17) to obtain a dimensionless equation:

The Eq. (13) is non-dimensionalized into:

Eqs. (18)-(20) are the main governing equations for the evolution of surfactant concentration, thickness and drainage velocity of liquid film under the excitation of low-frequency artificial seismic waves.

We assumed that the thickness of liquid film at the upper end and the lower end is constant:

The fluid at the top and bottom of the wire frame does not flow:

The dimensionless initial thickness of the foam liquid film is:

The initial concentration of dimensionless surfactant in the foam liquid film is:

Master Eqs. (18)-(20), boundary condition Eqs. (21)-(22) and initial condition Eqs. (23)-(24) are written in difference schemes. For the convenience of calculation, the thickness of the liquid film and the concentration of surfactant in the liquid film were determined at the full point, and the drainage velocity of the liquid film was determined at the half point. The implicit and explicit methods are used to solve the above-mentioned partial differential equations. In the process of solving, the one- dimensional region within 0 < x_D < 1 was selected according to the flow scale. Under the condition of ensuring the accuracy and shortening the calculation time, the calculation area was divided into 100 equal parts for calculation to obtain the dimensionless simulation results.

In order to verify the accuracy of the solution of the model, the acceleration in Eqs. (18)-(20) of the model is set as 0 (wave force is 0) to obtain the simplified model under low-frequency wave excitation (abbreviated as simplified model). Referring to the values of basic parameters in the literature model (Table 1), the relationship curves of dimensionless liquid film surfactant concentration, dimensionless liquid film thickness and dimensionless liquid film drainage distance under different dimensionless drainage time conditions in the simplified model were calculated respectively, and the results were compared with the calculation results of the model in References [8, 11] (referred to as the literature model) (shown in Fig. 2) (for the convenience of expression, "dimensionless" will be omitted later, unless otherwise specified, the physical quantities involved in the paper and figure are all dimensionless). It can be seen from Fig. 2 that the concentrations of surfactant and the thicknesses of liquid film calculated by the two models have the same distribution law, which verifies the correctness of the model.

According to the research results of Ye et al.^[11], a half-moon shaped area is formed at the top of the liquid film during the liquid film drainage process, and a spherical bulge is formed at the bottom due to the accumulation of liquid, and the middle part is the area with small curvature and nearly straight shape of the liquid film. With the increase of drainage time, the liquid film with the minimum thickness appears at the end of the half-moon region, and when the liquid film drainage time t_D=4×10^-3, a long area in the middle part of the liquid film has thickness very close to the minimum liquid film thickness, which corresponds to the black film area in the experiment. Therefore, the area close to the minimum liquid film thickness is called "black film", and Marangoni effect is the strongest at this time. Therefore, when t_D=4×10^-3, the foam liquid film drainage reaches the later stage of liquid drainage. The vibration acceleration was set at 0.5 times the acceleration of gravity, and the concentration distributions of surfactant in the foam liquid film under different vibration frequencies were calculated (shown in Fig. 3). It can be seen from Fig. 3 that the surfactant concentration in foam liquid film presents a trend of gradual increase from top to bottom. The difference between the maximum and minimum surfactant concentrations in foam liquid film is 64 without vibration. With vibration, the difference between the maximum and minimum surfactant concentrations in foam liquid film decreases significantly, and the difference first decreases and then increases with the increase of vibration frequency. When the vibration frequencies are 50 Hz and 60 Hz, the difference between the maximum and minimum value of surfactant concentration reaches the minimum (about 44), which is a reduction by 31.3% compared with the case without vibration. When the vibration frequency is close to the natural frequency of the object, the object is prone to generate resonance effect, and the vibration effect is the best^[18,19]. According to the above analysis, it can be seen that 50 Hz and 60 Hz are close to the natural frequency of the foam liquid film.

Fig. 4 shows the concentration distributions of surfactant in the foam liquid film at different times in the case with no vibration and the case at the vibration frequency of 50Hz (Note: the two data of t_D=8×10^-7and t_D=1×10^-8 coincide). With or without vibration, when 0<t_D≤8×10^-7, the surfactant concentration in the upper region of the liquid film gradually decreases and the surfactant concentration at the bottom increases with the increase of time. When 1×10^-4≤t_D≤4×10^-3, with the increase of time, the concentration of surfactant on the top of foam liquid membrane increases, and the concentration of surfactant in the lower liquid film decreases. This indicates that the surfactant molecules go upstream under the effect of Marangoni effect from the time of t_D=1×10^-4, making the difference between the maximum and minimum surfactant concentrations gradually decrease. Meanwhile, the existence of Marangoni effect delays the drainage velocity of the foam liquid film, and the stability of the foam liquid film is improved. When t_D=4×10^-3, the foam liquid film drainage tends stable, and the difference between the maximum and minimum of surfactant concentrations reaches the minimum.

By comparing the whole drainage processes under the vibration frequency of 50 Hz and with no vibration, we found that when t_D=1×10^-8 and t_D=8×10^-7, the surfactant distribution curves with vibration and no vibration coincide, which indicates that the low frequency vibration had no effect on the liquid film drainage in the early stage. With the drainage going on, the Marangoni effect appeared at t_D=1×10^-4, at this point the influence of low-frequency vibration on the Marangoni effect in the liquid film began to appear. When t_D=4×10^-3, the influence of low-frequency vibration on Marangoni effect reaches the maximum, and the difference between the maximum and minimum surfactant concentrations in the foam liquid film was smaller. In the later stage of liquid film drainage, low frequency vibration can improve the recovery rate of surfactant concentration in the foam liquid film, thereby enhancing the Marangoni effect and improving the stability of the foam liquid film.

Fig. 5 and Fig. 6 show the distributions of film thickness and liquid film drainage velocity at different positions in the case with no vibration and with vibration frequency of 50 Hz and 60 Hz at t_D=4×10^-3, respectively. It can be seen that when t_D=4×10^-3, compared with the case with no vibration, in the cases with vibration, the velocity of liquid film drainage and the bulge radius at the bottom of liquid film formed by the accumulation of liquid volume significantly reduce. In the cases with the vibration frequency of 50 Hz and 60 Hz, the drainage velocities of the liquid film, and the convex radii at the bottom are the same, which further demonstrates that low frequency vibration can change the distribution of liquid film thickness, slow down the drainage velocity, and improve the foam stability, which is consistent with the results of previous experiments and theoretical research^[15], and further confirms that the model is reliable.

Since the initial thickness of the liquid film is the same, compared with the case with no vibration, in the case with vibration, the radius of the bulge formed at the bottom of the liquid film is smaller. Therefore, in the direction of fluid drainage, when 0.65<x_D<0.80, the foam liquid film thickness is larger and the length of the minimum liquid film thickness is shorter in the case with vibration than the case without vibration. The reason is that the vibration delays the drainage velocity of the liquid film, so the thickness of the foam liquid film changes little in the same time, and it takes longer time to reach the length corresponding to the same minimum liquid film thickness under vibration conditions, in other words, vibration can improve the stability of foam. It can also be seen from Fig. 6 that compared with the case with no vibration, the drainage position corresponding to the maximum liquid drainage velocity of the liquid film under low-frequency vibration condition is on the upper part. This is because the position where the maximum liquid film drainage velocity occurs corresponds to the position at the end of the minimum liquid film thickness.

In the cases with the vibration frequencies of 50Hz and 60Hz, the concentrations of surfactant, thicknesses of the liquid film and drainage velocities in the foam liquid film are similar. Considering the actual situation on the site, 50Hz is selected as the best vibration frequency.

Under the optimal vibration frequency, the concentration distributions of surfactant in foam liquid film at different vibration accelerations at t_D=4×10^-3 were calculated (shown in Fig. 7). It can be seen that when the vibration acceleration is less than or equal to 0.8 times the gravity acceleration, the concentration of surfactant in the foam liquid film gradually increases from top to bottom, and remains unchanged after reaching a certain value. The difference between the maximum and minimum surfactant concentrations decreases with the increase of vibration acceleration, and is always lower than that in the case without vibration. This indicates that the greater the vibration acceleration, the higher the recovery rate of surfactant concentration in the foam liquid film will be, the stronger the Marangoni effect will be to delay the drainage of the foam liquid film, and the better the foam stabilizing performance of the Marangoni effect will be. When the vibration acceleration is equal to 3.0 times the gravity acceleration, the surfactant concentration on the top of the foam liquid film is much higher than that at the bottom of the liquid film, which is manifested as the surfactant flowing upward and accumulating at the top of the liquid film, that is contrary to the fact that the liquid film flows downward. Under the action of Marangoni effect, the surfactant can diffuse reversely, recover the local surfactant concentration of the liquid film to a certain extent, and delay the liquid film drainage. However, the Marangoni effect has a limited effect, which can only delay the drainage of liquid film, but cannot reverse the direction of drainage flow. Therefore, it isn’t the higher, the better the vibration acceleration.

Fig. 8 and Fig. 9 show the distributions of liquid film thickness and drainage velocity under different vibration accelerations when t_D=4×10^-3, respectively. It can be seen from the figure that when the vibration acceleration is less than 0.8 times of the gravity acceleration, the bulge radius formed by the liquid film drainage accumulation and the liquid film drainage velocity decrease with the increase of the vibration acceleration. When the vibration acceleration is equal to 3 times the acceleration of gravity, the liquid drainage direction of the foam liquid film is upward, and the maximum liquid film drainage speed is about 2 times that of the case without vibration. This leads to the reverse accumulation of the foam liquid on the top of the liquid film, which is obviously inconsistent with the fact.

The vibration acceleration, vibration frequency and vibration displacement meet the following relation:

It can be seen that for the same vibration equipment with given vibration displacement range, when the optimal vibration frequency is determined, its maximum vibration acceleration is also determined. The research of Li et al.^[14] showed that vibration acceleration too large or too small could inhibit foaming effect. Combined with the understanding of this paper, the optimal vibration acceleration is 0.5 times of gravity acceleration.

Under the optimal vibration parameters, by changing the initial concentration of surfactant from 100 to 1000, 1500 and 5000, respectively, the distributions of liquid film thickness at different initial surfactant concentrations at the drainage time of t_D=4×10^-3 (shown in Fig. 10) were calculated. It can be seen that under the same conditions, the radius of bulge at the bottom of the liquid film decreases with the increase of the initial surfactant concentration. This is because when the concentration of surfactant in the liquid film is low, the concentration difference of surfactant is relatively small, and Marangoni effect is relatively weak and difficult to maintain the stability of liquid membrane for a long time. When the initial concentration of surfactant increases, the concentration difference of surfactant in the liquid film increases correspondingly, and the Marangoni effect gradually enhances. The bulge formed at the bottom of liquid film with high initial surfactant concentration is relatively small in radius.

In addition, at different initial concentrations of surfactant, the vibration has different influence degrees on the bottom bulge radius of liquid film under the same vibration parameters (Table 2). With the increase of the initial concentration of surfactant in the liquid film, the difference between the radii of bulges at the bottom of the liquid film in the cases with and without vibration gradually decreases, which indicates that the smaller the initial concentration of surfactant is, the better the vibration effect is. This is because when the initial concentration of surfactant is low, the Marangoni effect in the liquid membrane is relatively weak, and has limited ability to delay the liquid film drainage speed, whereas vibration can greatly improve the effect of Marangoni effect.

By changing the initial surfactant concentration, the distributions of liquid film drainage velocity at different initial surfactant concentrations at t_D=4×10^-3 were calculated (Fig. 11). It can be seen that the drainage velocity of the liquid film decreases with the increase of the initial surfactant concentration in the liquid film in the case without vibration. Under the same initial concentration of surfactant, the difference between the drainage velocity of liquid film in the case without vibration and that with vibration decreases with the increase of initial surfactant concentration, indicating that the smaller the initial concentration of surfactant, the better the vibration effect.

It can also be seen in Fig. 11 that in the case without vibration, when x_D is about 0.65, the maximum liquid film drainage velocity at the initial surfactant concentration of 1000 is greater than that at the initial concentration of 100, which is abnormal. It is inferred through analysis that the liquid film with the initial surfactant concentration of 100 reaches the "black film" stage first, and the liquid film drainage speed is very small in this stage. When the initial surfactant concentration is 1000, the liquid film is thicker, and the maximum velocity appears near the bulge at the lower end of the liquid film. Compared with the gravity of the liquid film, the Marangoni effect has less influence, so the liquid film drainage speed is larger. The maximum value of the liquid film drainage velocity with the initial surfactant concentration of 1500 is the same as that of the liquid film with the initial concentration of 100, which indicates that the Marangoni effect gradually enhances and the liquid film drainage speed decreases at this surfactant concentration. When the initial surfactant concentration is further increased to 5000, Marangoni effect further enhances, and the liquid film drainage rate drops greatly.

Low frequency vibration can reduce the difference between the maximum and minimum surfactant concentrations in the foam liquid film at the late stage of liquid drainage, enhance the effect of Marangoni effect, and improve the stability of the foam liquid film. When the vibration frequency is close to the natural frequency of foam liquid film, the vibration effect is best, and the best vibration frequency is about 50 Hz.

The higher the vibration acceleration, the faster the recovery rate of surfactant concentration in the foam liquid film, the stronger the Marangoni effect is to postpone the drainage capacity of the foam liquid film, and the better the foam stabilizing performance is. But it is not the faster the vibration acceleration, the better, and the best vibration acceleration is about 0.5 times of gravity acceleration.

Reasonable vibration parameters can greatly improve the effect of Marangoni effect. The smaller the initial concentration of surfactant, the better the vibration works in enhancing on Marangoni effect.

a—vibration acceleration, m/s²;

A—separation pressure strength coefficient, Pa;

A_D—dimensionless separation pressure coefficient;

B_D—the dimensionless modified Bond number, indicating the ratio of gravity to capillary force;

C_D—dimensionless capillary number;

C_S—surfactant concentration, mol/L;

C_S0—initial surfactant concentration, mol/L;

C_SD—dimensionless surfactant concentration;

C_S0D—dimensionless initial surfactant concentration;

$C_{\mathrm{SD}}\left(0, t_{\mathrm{D}}\right)$—dimensionless surfactant concentration at the upper end of the liquid film;

$C_{\mathrm{SD}}\left(1, t_{\mathrm{D}}\right)$—dimensionless surfactant concentration at the lower end of the liquid film;

$C_{\mathrm{SD}}\left(X_{\mathrm{D}}, 0\right)$—dimensionless surfactant concentration at initial time of liquid film;

D—diffusion coefficient, m²/s;

D_D—dimensionless diffusion coefficient;

f—vibration frequency, Hz;

f_D—dimensionless vibration frequency;

F_b—vibrational force, N/m³;

g—gravity acceleration, m/s²;

h—thickness of liquid film, m;

h₀—initial thickness of liquid film, m;

h_D—dimensionless liquid film thickness;

h_eq—equilibrium liquid film thickness, m;

$h_{\mathrm{D}}\left(0, t_{\mathrm{D}}\right)$—dimensionless thickness of upper end of liquid film;

$h_{\mathrm{D}}\left(1, t_{\mathrm{D}}\right)$—dimensionless thickness at the bottom of liquid film;

$h_{\mathrm{D}}\left(X_{\mathrm{D}}, 0\right)$—dimensionless thickness of liquid film at initial time;

K—coefficient related to surfactant concentration, N•L/(mol•m);

L—characteristic length of liquid film, m;

p—pressure on liquid film, Pa;

p_D—dimensionless pressure on liquid film;

p_f—characteristic pressure, Pa;

p_se—separation pressure, Pa;

p_seD—dimensionless separation pressure;

p_S—surface pressure of liquid film, Pa;

p_SD—dimensionless liquid film surface pressure;

Q_0D—dimensionless slip flow;

Q_1D—dimensionless shear flow;

t—drainage time, s;

t_D—dimensionless drainage time;

T—characteristic time, s;

u—liquid drainage velocity of foam liquid film, m/s;

u₀—slip velocity, m/s;

u_0D—dimensionless slip velocity;

u₁—shear velocity, m/s;

u_1D—dimensionless shear velocity;

$u_{1 \mathrm{D}}^{(0)}$—the first term of series expansion of dimensionless shear velocity;

U₀—characteristic slip velocity, m/s;

U₁—characteristic shear velocity, m/s;

v—velocity of foam liquid film thickness, m/s;

V—characteristic velocity of foam liquid film thickness, m/s;

v_D—dimensionless velocity of foam liquid film thickness;

x—liquid drainage distance of foam liquid film, m;

x_D—dimensionless liquid film drainage distance;

y—liquid drainage distance perpendicular to x, m;

y_D—dimensionless liquid film drainage distance perpendicular to x;

z—vibration displacement, m;

α—attenuation coefficient, s^-1;

ε—ratio of film initial thickness to film characteristic length, dimensionless;

κ—correction factor, dimensionless;

λ—surface curvature of liquid film, m^-1;

μ—viscosity of foam solution, Pa•s;

ρ—density of foaming agent solution, kg/m³;

σ—surface tension, N/m;

σ₀—initial surface tension, N/m;

τ—shear stress, Pa;

τ_D—dimensionless shear stress.