Numerical Simulation of the Enrichment of Chemotactic Bacteria in Oil-Water Two-Phase Transfer Fields of Heterogeneous Porous Media

Abstract

Oil pollution in soil-groundwater systems is difficult to remove, and a large amount of residual oil is trapped in the low permeable layer of the heterogeneous aquifer. Aromatic hydrocarbons in oil have high toxicity and low solubility in water, which are harmful to the ecological environment. Chemotactic degrading bacteria can perceive the concentration gradient of non-aqueous phase liquid (NAPL) pollutants in the groundwater environment, and enrich and proliferate around the pollutants, thus achieving a more efficient and thorough remediation effect. However, the existing theoretical models are relatively simple. The physical fields of oil–water two-phase flow and oil-phase solute convection and diffusion in water are not coupled, which further restricts the accuracy of studies on bacterial chemotaxis to NAPL. In this study, geometric models based on the actual microfluidic experimental study were constructed. Based on the phase field model, diffusion convection equation and chemotaxis velocity equation, the effects of heterogeneity of porous media, wall wettability and groundwater flow rate on the residual oil and the concentration distribution of chemotaxis bacteria were studied. Under all of the simulation conditions, the residual oil in the high permeable area was significantly lower than that in the low permeable area, and the wall hydrophilicity enhanced the water flooding effect. Chemotactic bacteria could react to the concentration gradient of pollutants dissolved into water in the oil phase, and enrich near the oil–water interface with high concentration of NAPL, and the density of chemotactic bacteria at the oil–water interface can be up to 1.8–2 times higher than that in the water phase at flow rates from 1.13 to 6.78 m/d.

1. Introduction

By 2020, the global oil production was about 4 billion tons per year, and about 8 million tons of crude oil entered the soil-water environment as a pollutant every year. Coastal oil pollution is an important part of soil-water pollution. Oil pollution resulted in reduced coastal vegetation and coastal organisms, increased erosion of swamp marginal habitats, disrupted nutrient cycling and water filtration, and the other regulatory functions of swamps, causing serious damage to coastal ecosystems [1].

As a typical non-aqueous phase liquid (NAPL), leaked oil has a complex composition, low solubility in water, strong binding ability with soil organic matter, strong anti-desorption ability and is difficult to remove from the soil aquifer [2]. After entering the soil, it blocks the pore structure of soil particles, changes the soil physical properties such as soil texture, compaction degree, structure state and the concentration and content of soil minerals and heavy metals, which seriously affect the functions of soil and the underground ecological environment [3]. Many hydrocarbon pollutants, especially polycyclic aromatic hydrocarbons (PAHs), also inhibit plant growth, damage the diversity and integrity of the soil microbial community, and disrupt or even destroy the regular soil micro-ecological environment. Direct or indirect contact with organisms may cause serious harm to biological health. Nowadays, petroleum-contaminated soil remediation technologies can be divided into three categories, according to the different disciplines involved: physical; chemical; and biological remediation technologies. However, the treatment process of physical and chemical remediation technology is complex, and the cost is high. It is easy to produce secondary pollution, and cannot fundamentally remove pollutants. It is usually only suitable for repairing small areas and high concentrations of petroleum-contaminated soil. For the remediation of large-scale and low-concentration petroleum-contaminated soil, compared with various physical and chemical decontamination methods, bioremediation technology has the advantages of high efficiency, economy and environmental protection [4]. Bacterial chemotaxis refers to a microbial characteristic of bacteria [5,6], which can respond to the gradient of chemical concentration in water and migrate to areas conducive to its survival and growth.

Many studies showed that bacteria have chemotaxis to many petroleum hydrocarbon pollutants [7]. Researchers observed the chemotaxis migration (chemotaxis) of chemotactic bacteria to the contaminated layer by designing heterogeneous sand columns or sand boxes [8,9,10,11], but the black box effect reduced the repeatability of the experiment, bringing great difficulties to the quantitative results. Datta et al. carried out innovative research on mesoscopic scale (Darcy scale), which laid the foundation for visualization of chemotactic bacteria on mesoscopic scale [12,13]; some experiments used microfluidic chips to study the remediation of non-aqueous pollution in heterogeneous soil at the micro-scale [14,15,16,17]. Other experiments showed that exercise-type bacteria had chemotaxis towards petroleum hydrocarbon pollutants that they could degrade. Most bacteria have sensory systems that can help them detect and respond to petroleum hydrocarbon pollutants. The chemotaxis of bacteria and their ability to degrade petroleum hydrocarbons enable them to contact directly with soil pollutants, thereby accelerating the degradation of petroleum hydrocarbons in these areas. Many of the microbes living in oil reservoirs consume hydrocarbons. In order to do this, they must break down the long alkyl chains that are present in heavy crude oil. By degrading these chains, bacteria increase the amount of valuable, light crude [18]. Some microbial oil displacement bacteria also have chemotaxis [19], so studying and understanding bacterial chemotaxis has a strong practical significance for oil pollution remediation and enhanced oil recovery (EOR) [20].

Many experimental [21] results showed that the chemotaxis of bacteria to petroleum hydrocarbon pollutants [22] can indeed strengthen the bioremediation of petroleum contaminated areas [23]. Therefore, in order to achieve a better remediation effect for petroleum-contaminated soil, it is very important to conduct theoretical research on the chemotaxis and migration of bacteria to petroleum hydrocarbon pollutants. In previous studies, the oil–water two-phase flow is often simplified as a single-phase water flow, and then the chemotaxis enrichment effect of bacteria in single-phase convection diffusion field is studied. The numerical model is quite different from the actual oil–water two-phase flow; the numerical simulation of two-phase flow ignores the concentration gradient of oil phase components dissolved in water at the oil–water interface, which hinders the study of bacterial chemotaxis under the condition of oil–water two-phase flow. In this paper, the experimental phenomena (chemotaxis of bacteria to oil–water interface of high concentration petroleum hydrocarbon pollutants) observed and summarized by previous people are taken as the research background. Combined with the relevant mathematical models, the numerical simulation of chemotaxis bacteria migration in the oil–water two-phase diffusion flow field is carried out. The innovation of this study is to couple the three processes of oil–water two-phase flow, the diffusion and convection of pollutants in oil to water and the migration of chemotaxis bacteria, and to study the phenomenon of convection, dissolution and diffusion of oil-phase pollutants in water and the migration of chemotaxis bacteria in water during water flooding.

This study provides a novel and improved numerical simulation method for the migration of chemotactic bacteria in the oil–water two-phase transfer field, provides simulation verification and support for the experimental phenomena of migration and aggregation of chemotactic bacteria in oil-contaminated areas, and provides a theoretical basis for the development of the bioremediation of oil-contaminated soil, based on bacterial chemotaxis.

2. Materials and Methods

2.1. Construction of Geometric Model

The finite-element solver COMSOL Multiphysics was used to accomplish the numerical modelling and simulation. Referring to the heterogeneous microfluidic chip used in the laboratory, the geometric model designed and constructed is shown in Figure 1. The specific size of the chip is shown in

Table 1. Summary of micromodel properties.

Chip	Permeability	Porosity	k/(μm2)	khigh/klow	kc	Atrix/(mm)	Throat/(mm)
(1)	Low	0.282	2.69	4.45	5.52	0.2	0.005
	High	0.476	11.98			1.0	0.300
(2)	Low	0.282	2.69	2.25	3.71	0.2	0.005
	High	0.275	6.06			1.0	0.100
(3)	Low	0.400	0.95	6.37	2.44	0.1	0.005
	High	0.275	6.06			1.0	0.100
(4)	Low	0.400	0.95	12.61	3.89	0.1	0.005
	High	0.476	11.98			1.0	0.300

.

Since the initial chip was too complex, it still failed to converge after ten days of simulation on the workstation. In order to reduce the amount of calculation, make the results more precise and better solve the problem, a part of the chip (in the red box) is intercepted in this simulation to study. The final chip is shown in Figure 2. In Figure 2, the geometric model diagram is divided into two regions, one region in the blue box and the other two regions.

Under the initial conditions, under the initial conditions, the two regions of the model are filled with oil phase, toluene is dissolved in the oil phase, and the water phase carries chemotactic bacteria in the region 1. The leftmost boundary of the model is the entrance, and the rightmost boundary is the exit. The water phase carrying chemotactic bacteria enters the displacing oil phase from the entrance.

2.2. Phase Field Model

When simulating the pore scale of the fluid flow in porous media, a method that can handle complex geometric and topological changes is needed.

The physical model of the phase field method is a diffusion interface method, based on density functional theory. The description of the interface uses thermodynamic principles. The origin of the phase field model can be traced back to the density gradient theory proposed by Van der Waals and Reynolds. In the phase field model, the interface has a finite thickness, and a diffusive interface of finite thickness separates two immiscible fluids, in contrast to classical hydrodynamics, which describes the interface between the two fluids as a free boundary, a key difference. Simulation of multiphase flow in a micropore-scale system with a phase field model achieves the best balance of thermodynamic rigor and computational efficiency.

Since the phase-field model can simulate more accurate and complex two-phase flow in porous media at the pore scale, more and more researchers choose the phase-field model for the microscopic study of the two-phase flow in porous media. Maaref used the phase field model to numerically simulate the oil–water two-phase flow in the microscopic model of porous media [24]. The calculated results under different wetting states, heterogeneous conditions and fluid viscosity were compared with the experimental results, and good consistency was found [24]. Amiri studied the non-isothermal oil–water two-phase flow in porous media at pore scale by coupling the phase field equation with the heat transfer equation, and discussed the influence of viscous force, capillary force, matrix wettability and heterogeneity on displacement effect [25]. The phase field model is more consistent with the experimental situation in the simulation process, which makes up for the deficiency of Darcy’s law in the application of porous media. The control equations of the phase field model mainly include the Continuity equation, Navier–Stokes equation and Cahn–Hilliard equation. The velocity field of two-phase flow can be obtained by solving this set of equations. The nomenclature section to the paper is shown in

Table 2. Nomenclature section to the paper.

Parameter Notation	Parameter Definition	Unit
u	Fluid velocity	m/s
ρ	The mixed fluid density in the two-phase transition Interface	kg/m3
t	Time	s
p	Pressure	Pa
I	Unit matrix	dimensionless
μ	Mixed fluid viscosity	Pa·s
T	Temperature	K
F	Surface tension	N
g	Gravity acceleration	kg/s2
ε	Interface thickness control parameter	m
λ	Mixed energy density	N
φ	Phase field variables	dimensionless
ψ	Phase field covariates	dimensionless
ρ1	Density of fluid 1	kg/m3
ρ2	Density of fluid 2	kg/m3
μ1	The viscosity of fluid 1	Pa·s
μ2	The viscosity of fluid 2	Pa·s
Vf2	Volume fraction of fluid 2	dimensionless
γ	Migration rate	m3·s/kg
σ	Surface tension coefficient	N/m
a	Chemoattractant concentration	mol/m3
De,a	Chemical attractant effective diffusion coefficient	m2/s
vx	Two-dimensional phase field model to solve the Fluid transverse velocity	m/s
vy	The longitudinal velocity of the fluid solved by the two-dimensional phase field model	m/s
Da	Diffusion coefficient of chemical attractant	m2/s
εp	Porosity	dimensionless
b	Bacterial density	mol/m3
vx	Two-dimensional phase field model to solve the fluid transverse velocity	m/s
vy	Two-dimensional phase field model to solve the fluid longitudinal velocity	m/s
vChx	Horizontal chemotaxis rate of bacteria	m/s
vChy	The vertical chemotaxis rate of bacteria	m/s
Db	Random motion coefficient of bacteria	m2/s
vb	Bacterial swimming speed	m/s
Kc	Chemokine receptor constant	mol/m3

.

2.2.1. Continuity Equation

For incompressible stable flow, the Continuity equation is:

$$\nabla \cdot \left(\mathit{u}\right)=0$$

(1)

u—Fuid velocity, m/s.

2.2.2. Navier–Stokes Equations

The Navier–Stokes equation replaces Darcy’s law to describe the fluid flow in porous media and calculates the average linear velocity of fluid, so as to obtain accurate results on the pore scale, and describes the mass transfer and momentum transfer of the two-phase fluid. Surface tension is added to the model to explain the capillary effect [26]. The Navier–Stokes equation is:

$$\rho \frac{\partial \mathit{u}}{\partial t}+\rho (\mathit{u}\cdot \nabla )\mathit{u}=\nabla [-p\cdot \mathit{I}+\mu (\nabla \mathit{u}+{(\nabla \mathit{u})}^T)]+\mathit{F}+\rho g$$

(2)

Because the model is in a two-dimensional horizontal plane, ignoring the gravity, that is, ρg = 0, the formula is obtained:

$$\rho \frac{\partial \mathit{u}}{\partial t}+\rho (\mathit{u}\cdot \nabla )\mathit{u}=\nabla [-p\cdot \mathit{I}+\mu (\nabla \mathit{u}+{(\nabla \mathit{u})}^T)]+\mathit{F}$$

(3)

• ρ—The mixed fluid density in the two-phase transition interface, kg/m³;
• u—Fluid velocity, m/s;
• t—Time, s;
• p—Pressure, Pa;
• I—Unit matrix, dimensionless;
• μ—Mixed fluid viscosity, Pa·s;
• T—Temperature, K;
• F—Surface tension, N;
• g—Gravity acceleration, kg/s².

F can be expressed by Formula (4):

$$\mathit{F}=G\nabla \phi ,$$

(4)

G is a chemical potential, J/m³, expressed by Formula (5):

$$G=\lambda \left[-{\nabla }^2\cdot \phi +\frac{\phi \left({\phi }^2-1\right)}{{\epsilon }^2}\right]=\frac{\lambda }{{\epsilon }^2}\cdot \psi$$

(5)

• ε—Interface thickness control parameter, m;
• λ—Mixed energy density, N;
• φ—Phase field variables, dimensionless;
• ψ—Phase field covariates, dimensionless.

Fluid viscosity μ and mixed fluid density ρ in the two-phase transition interface are expressed as follows:

$$\{\begin{array}{c}\rho ={\rho }_1+\left({\rho }_2-{\rho }_1\right)V_{f2}\\ \mu ={\mu }_1+\left({\mu }_2-{\mu }_1\right)V_{f2}\end{array}$$

(6)

• ρ₁—Density of fluid 1, kg/m³;
• ρ₂—Density of fluid 2, kg/m³;
• μ₁—The viscosity of fluid 1, Pa·s;
• μ₂—The viscosity of fluid 2, Pa·s;
• V_f2—Volume fraction of fluid 2, dimensionless.

  $$\{\begin{array}{c}{V}_{f1}=\frac{1-\phi }{2}\\ V_{f2}=\frac{1+\phi }{2}\end{array},$$

  (7)

The range of phase field variable φ is [−1,1]. In the homogeneous phase region, the value of φ is −1 or 1. When φ = 1, the phase region is pure fluid 2, and when φ = −1, the phase region is pure fluid 1. In the transition zone between fluid 1 and fluid 2, when φ changes uniformly, −1 < φ < 1. In this oil–water two-phase flow simulation, the oil phase as the displaced phase is fluid 1, namely φ = −1, and the water phase as the displaced phase is fluid 2, namely φ = 1.

2.2.3. Cahn–Hilliard Equation

The Cahn–Hilliard equation is a phase field equation commonly used to reflect the evolution of conservative field variables that reflect the change of composition [27]. It can be used to track the diffusion interface of oil–water two phases, considering the different effects of three scales: macroscopic fluid mechanics; supramolecular or intermolecular interactions; and mesoscopic interface morphology. The diffusion and deformation of the oil–water two-phase interface caused by macroscopic fluid dynamics and intermolecular forces can be well described [17]. The Cahn–Hilliard equation is:

$$\{\begin{array}{c}\frac{\partial \phi }{\partial t}+\mathit{u}\cdot \nabla \phi =\nabla \cdot \frac{\gamma \lambda }{{\epsilon }^2}\nabla \psi \\ \psi =-\nabla \cdot {\epsilon }^2\nabla \phi +\left({\phi }^2-1\right)\phi +\frac{{\epsilon }^2}{\lambda }\frac{\partial f}{\partial \phi }\end{array}$$

(8)

f is the external free energy and ∂f/∂φ is the derivative of the external free energy with respect to φ. ∂f/∂φ takes 0 in general, and the formula is obtained:

$$\{\begin{array}{c}\frac{\partial \phi }{\partial t}+\mathit{u}\cdot \nabla \phi =\nabla \cdot \frac{\gamma \lambda }{{\epsilon }^2}\nabla \psi \\ \psi =-\nabla \cdot {\epsilon }^2\nabla \phi +\left({\phi }^2-1\right)\phi \end{array}$$

(9)

• φ—Phase field variables, dimensionless;
• t—Time, s;
• u—Fluid velocity, m/s;
• γ—Migration rate, m³·s/kg;
• λ—Mixed energy density, N;
• ψ—Phase field variables, dimensionless;
• ε—Interface thickness control parameter, m.

The mobility γ can be expressed as:

$$\gamma ={\gamma }_c{\epsilon }^2,$$

(10)

γ_c—Migration adjustment parameters, m·s/kg.

The mixed energy density λ can be expressed as:

$$\lambda =\frac{3\epsilon \sigma }{\sqrt{8}},$$

(11)

σ—Surface tension coefficient, N/m.

2.3. Convection—Diffusion Equation of Chemoattractant

Based on the oil–water two-phase flow model, it is necessary to establish the convection–diffusion equation of chemical attractants to simulate the diffusion of pollutants (toluene) in oil phase to water phase.

Convection–diffusion equations for chemo-attractants describe the dissolution of bacterial chemotaxis (e.g., toluene) into water [28]. The concentration distribution and the concentration gradient of chemical attractants can be obtained by solving this equation.

The two-dimensional plane chemoattractant convection–diffusion equation used in this simulation is:

$$\frac{\partial a}{\partial t}=D_{e,a}\frac{{\partial }^{2}a}{\partial x^2}+D_{e,a}\frac{{\partial }^{2}a}{\partial y^2}-\frac{v_x\partial a}{\partial x}-\frac{v_y\partial a}{\partial y},$$

(12)

• a—Chemoattractant concentration, mol/m³;
• t—Time, s;
• D_e,a—Chemical attractant effective diffusion coefficient, m²/s;
• v_x—Two-dimensional phase field model to solve the fluid transverse velocity, m/s;
• v_y—The longitudinal velocity of the fluid solved by the two-dimensional phase field model, m/s.

The effective coefficient of chemical attractants D_e,a is expressed as:

$$D_{e,a}={\epsilon }_p{D}_a,$$

(13)

• D_a—Diffusion coefficient of chemical attractant, m²/s;
• ε_p—Porosity, dimensionless.

In this simulation, the chemoattractant was toluene in the oil phase.

2.4. Convection–Diffusion Equation and Chemotaxis Velocity Equation of Bacteria

On the basis of the oil–water two-phase transfer field model, it is necessary to couple the convection–diffusion equation and the chemotaxis velocity equation of bacteria to realize the migration simulation of chemotaxis bacteria in the oil–water two-phase diffusion flow field.

The convection–diffusion equation of bacteria was rewritten by Olson et al. [29]. The chemotaxis rate equation of bacteria describes the advection migration of bacteria to the concentration gradient of chemical attractants, which is caused by chemotaxis. The equation was proposed by Chen et al. [21]. The two equations were solved and the density distribution of bacteria was obtained. Wang et al. used these two equations to describe the transport and migration of chemotactic bacteria when studying the effect of residual NAPL on the retention of chemotactic bacteria in pore networks [16].

On the two-dimensional plane, the convection–diffusion equation of bacteria is:

$$\frac{\partial b}{\partial t}=D_{e,b}\frac{{\partial }^{2}b}{\partial x^2}+D_{e,b}\frac{{\partial }^{2}b}{\partial y^2}-\frac{v_x\partial b}{\partial x}-\frac{v_y\partial b}{\partial y}-\frac{v_{Chx}\partial b}{\partial x}-\frac{v_{Chy}\partial b}{\partial y},$$

(14)

• b—Bacterial density, mol/m³;
• t—Time, s;
• D_e,b—Effective random movement coefficient of bacteria, m²/s;
• v_x—Two-dimensional phase field model to solve the fluid transverse velocity, m/s;
• v_y—Two-dimensional phase field model to solve the fluid longitudinal velocity, m/s;
• v_Chx—Horizontal chemotaxis rate of bacteria, m/s;
• v_Chy—The vertical chemotaxis rate of bacteria, m/s.

D_e,b—is the random motion coefficient of bacteria, which can be expressed as:

$$D_{e,b}={\epsilon }_p{D}_b,$$

(15)

• D_b—Random motion coefficient of bacteria, m²/s;
• ε_p—Calculation porosity of pore network, dimensionless.

In the two-dimensional plane, the bacterial chemotaxis rate equation is:

$$\{\begin{array}{c}{v}_{Chx}=\frac{2}{3}{v}_b\tanh \left(\frac{{\chi }_0}{2v_b}\frac{K_c}{{\left(K_c+a\right)}^2}\frac{\partial a}{\partial x}\right)\\ v_{Chy}=\frac{2}{3}{v}_b\tanh \left(\frac{{\chi }_0}{2v_b}\frac{K_c}{{\left(K_c+a\right)}^2}\frac{\partial a}{\partial y}\right)\end{array},$$

(16)

• v_b—Bacterial swimming speed, m/s;
• χ₀—The chemotactic sensitivity coefficient, m²/s;
• K_c—Chemokine receptor constant, mol/m³;
• a—Chemoattractant concentration, mol/m³.

In this simulation, chemotactic bacteria initially existed in the aqueous phase.

In this study, we first needed to use the phase field model to simulate the oil–water two-phase flow and obtain the velocity field of the two-phase flow. Secondly, on the basis of the oil–water two-phase flow model, the convection–diffusion equation of chemical attractants was established to simulate the transfer field of the oil–water two-phase flow. Finally, on the basis of the oil–water two-phase transfer field model, the convection–diffusion equation of bacteria and the bacterial chemotaxis velocity equation were established to simulate the enrichment of chemotaxis bacteria in the oil–water two-phase transfer field.

2.5. Establishment of the Model

The mathematical equations needed for this simulation are mentioned above. The simulation can be realized by using the interface of ‘laminar two-phase flow, phase field’ in the fluid flow module of COMSOL Multiphysics and ‘dilute material transfer’ in the chemical material transfer module.

2.5.1. ‘Laminar Two-Phase Flow, Phase Field’ Interface

This simulation realizes the simulation of oil–water two-phase flow and the solution of two-phase velocity field through this interface.

In the ‘laminar’ interface, the reference pressure level is 1 atm, and the reference temperature is 293.15 K. In the setting of the initial value 1, the velocity field u_x = u_y = 0, the leftmost boundary of the model is the inlet, the rightmost boundary is the outlet, and its boundary is the wall. The wall condition is set to slip, and the water phase velocity is set at the inlet. The normal inflow velocity is set to 1.13, 2.26, 4.52 and 6.78 m/d according to the typical groundwater flow velocity (1⁻¹⁰ m/d). Check the exit to suppress reflux. In the ‘phase field’ interface, the oil phase of the displaced phase is fluid 1, and the water phase of the displaced phase is fluid 2. In order to ensure that the simulation is easy to converge, in the displacement process, the oil phase (fluid 1 is (φ = −1)) initially exists in the region 2, and the water phase (fluid 2 is (φ = 1)) initially exists in the region 1. The leftmost boundary is the entrance. The phase field condition at the entrance is fluid 2 (φ = 1), which means that the entrance is filled with the water phase of the displaced phase, and the rightmost boundary is the exit. The wall, except for the inlet boundary and the outlet boundary, is a wetting wall, and the contact angle input different values, indicating the different wetting degree of the wall. In the phase field model, the interface thickness control parameter ε, the migration adjustment parameter γ_c and ∂f/∂φ are input. The interface thickness control parameter ε is the capillary width that controls the change of interface thickness. In COSMSOL Multiphysics, in order to balance the convergence and calculation speed, the value of ε is set as ε = h_max/2, and h_max is the maximum unit size of the mesh. In this simulation, the maximum grids in different models are different, so ε varies with the specific situation. When the migration adjustment parameter γ_c = 1 m·s/kg, the simulation results show less volume shrinkage and more realistic physical results, so the migration adjustment parameter γ_c is 1 m·s/kg in this simulation. For the reciprocal of ∂f/∂φ, according to the content of the Cahn–Hilliard equation, ∂f/∂φ = 0 J/m³. In the ‘two-phase laminar flow, phase field’ interface, the material of fluid 1 is Engine oil. The material of fluid 2 was Water, and the surface tension coefficient was set to 0.0006 N/m.

2.5.2. ‘Rare Material Transfer 1’ Interface

This interface controls the diffusion and convection of toluene in oil to water. The concentration distribution and concentration gradient of toluene can be obtained after solution.

In this simulation, the chemical attractant (toluene) value of Wang et al., in the study of residual NAPL pollutant-enhanced chemotaxis experiment bacteria in the pore grid, was taken as the physical parameter value required by toluene in the oil phase in the diffusion convection in this simulation as a reference [19]. In the transfer property, the velocity field is selected by the convective velocity field, indicating that the velocity field is the transverse velocity and longitudinal velocity of the fluid solved by the interface of ‘laminar two-phase flow and phase field’. The diffusion coefficient selects the user—defined and enters the statement ‘if (pf. Vf1 > 0, D_ea, D_w)’. It indicates that the diffusion coefficients of toluene in oil and water phases are different, and the diffusion coefficient of toluene in water phase is D_ea, and that in oil is D_w. Two regions were selected in the initial value 1, and the concentration was set to 5.65 mol/m³, indicating that the initial toluene concentration in the oil phase was 5.65 mol/m³. The leftmost boundary was selected as the inflow boundary, and the concentration was 0 mol/m³. Region 1 was selected in the initial value 2, and the concentration was 0 mol/m³, indicating that the concentration of toluene in the water phase of the displacement phase was 0 mol/m³. The outflow boundary selects the rightmost boundary. In the absence of flux, the boundary is selected to include all of the remaining walls except the inlet and outlet.

2.5.3. ‘Rare Material Transfer 2’ Interface

The transport process of chemotactic bacteria in oil–water two-phase transfer field is described by ‘dilute material transfer 2’ interface, and the density distribution of chemotactic bacteria can be obtained after solving.

We take the values of chemotactic bacteria in Wang et al. as a reference for the physical parameters needed to simulate the transport and migration of chemotactic bacteria in the transfer field. In the transfer property, the convection velocity field selects the user definition and enters u + (2 × v_b/3) × tanh(χ₀ × K_c × c_x/(2 × v_b × (K_c + c)²)) in the x-direction column and v + (2 × v_b/3) × tanh(χ₀ × K_c × c_y/(2 × v_b × (K_c + c)²)) in the y-direction column. The chemotactic bacterial swimming velocity v_b was 44 μm/s, the values of the effective random motion coefficients for bacteria in the different models are shown in

Table 3. Effective random motion coefficient of bacteria in different models.

Chips	Chip 1	Chip 2	Chip 3	Chip 4
Effective Random Motion Coefficient of Bacteria (m2/s)	4.927 × 10−10	3.62 × 10−10	4.3875 × 10−10	5.694 × 10−10

. the chemotactic sensitivity coefficient χ₀ was 1.8 × 10⁻⁸ m²/s, and the chemotactic factor receptor constant K_c was 1 mol/m³; where c is the toluene concentration calculated in the ‘dilute transfer 1’ interface, c_x is the transverse toluene concentration gradient and c_y is the longitudinal toluene concentration gradient.

In the initial value 1, region 2 was selected, and the concentration was set to 0 mol/m³, indicating that there was no chemotactic bacteria in the initial oil phase. The leftmost boundary was the inlet, and the concentration was 1 mol/m³. In the initial value 2, region 1 was selected, and the concentration was 1 mol/m³, indicating that the concentration of chemotactic bacteria in the water phase of the displacement phase was 1 mol/m³. The rightmost boundary is the exit. In the absence of flux, the boundary is selected to include all of the walls except the inlet and outlet.

So far, the geometric model is constructed by COMSOL, the boundary conditions and physical parameters are set up, the mesh is divided and the solver is set up. The numerical model of oil–water two-phase flow, convection diffusion and bacterial chemotaxis is obtained, which lays the foundation for the next step.

3. Results and Discussion

3.1. Distribution of Residual Oil in Heterogeneous Porous Media

3.1.1. Velocity Distribution

When the simulation is carried out to 1 min, the velocity field distribution of the four chip models under quasi-steady state conditions can be obtained, as shown in Figure 3.

It can be seen from Figure 3 that when the inlet velocity is 6.68 m/d, the flow velocity in the low permeable area of the model is almost 0 m/s under the quasi-steady state with different simulation chips, and there is a dominant channel in the high permeable area. In the dominant channel of chip 1, the faster flow velocity region exists in the throat between the particles in the hypertonic zone and the boundary of the particles, which is about 1 × 10⁻³~3 × 10⁻³ m/s. The lower flow velocity region exists in the entrance of the hypertonic zone, which is about 6 × 10⁻⁴~7 × 10⁻⁴ m/s. The velocity distribution of chip 2 is similar to that of chip 1. In the dominant channel, the flow rate is about 3 × 10⁻³~5 × 10⁻³ m/s in the region with a large flow rate, and 1 × 10⁻³–2 × 10⁻³ m/s in the region with a low flow rate. In the dominant channel of chip 3, the flow rate in the region with a high flow rate is about 4 × 10⁻³~6 × 10⁻³ m/s, and the flow rate in the region with a low flow rate is about 1.5 × 10⁻³~2 × 10⁻³ m/s. The high velocity region in the dominant channel of chip 4 is about 1.2 × 10⁻³~2.5 × 10⁻³ m/s.

3.1.2. Effect of Heterogeneous Porous Media on Residual Oil

The simulation shows that heterogeneity will affect the distribution of residual oil in the oil displacement process. Figure 4 show the changes in the oil volume fraction in different models in the oil displacement process. In all of the models, as shown in Table 1, the ratio of permeability in high and low permeable areas of chip 4 is the largest (k_{high permeable}/k_{low permeable} = 12.61), and the ratio of permeability in chip 2 is the smallest (k_{high permeable}/k_{low permeable} = 2.25). In addition, the ratio of permeability in chip 1 is k_{high permeable}/k_{low permeable} = 4.45, and the ratio of permeability in chip 3 is k_{high permeable}/k_{low permeable} = 6.37. Chip 1 is consistent with chip 4 (high permeable), chip 2 is consistent with chip 3 (low permeable); chip 1 is consistent with chip 2 (high permeable), chip 3 is consistent with chip 4 (low permeable).

Chip 1 is composed of large-size, low permeable zone and large throat, high permeable zone, and the permeability ratio of the high permeable zone to the low permeable zone is 4.45. The initial water flow advances smoothly in the upstream of the high permeable zone, and then the water flow evolves along a certain dominant channel. At 0.2 min, the model reaches a quasi-steady state, and the water flow breaks through the outlet to form a dominant channel. In chip 1, the water flow does not break through the low permeable zone.

Chip 2 is composed of a large particle size, low permeable zone and a small throat, high permeable zone. The ratio of permeability of the high permeable zone to the low permeable zone is 2.25. The chip 2 is the chip with the smallest permeability ratio in the four groups of models. Compared with chip 1, the time for chip 2 to form the dominant channel is slower than that of chip 1, and there are still residual oil droplets in the high permeable zone after 1 min. In the later stage, the water phase at the front end of the inlet tends to migrate to the low permeable zone.

Chip 3 is composed of a small particle size, low permeable zone and a small throat, high permeable zone. The ratio of permeability of high permeable zone to low permeable zone is 6.73. At 0.2 min, the flow begins to enter the front of the low permeable zone, and at the same time, the dominant channel is formed in the high permeable zone. When the breakthrough time is faster than that of chip 1 and chip 2, 1 min, there are residual oil droplets in the high permeable zone, and the water phase has entered the low permeable zone.

Chip 4 model is composed of a large throat, high permeable area and a small particle size, low permeable area. The ratio of permeability of the chip 4 high permeable area to low permeable area is 12.61, which is the largest in the four groups of models. The predominant channel area distribution of chip 1 and chip 2 is similar, the dominant channel formation time is greater than chip 3, 1 min after the water into the low permeable area.

Figure 5 is the curve of residual oil in high and low permeable zones in heterogeneous porous media. From the figure, the residual oil in low permeable zone is much higher than that in the high permeable zone under arbitrary simulation conditions. From the simulation results, the residual oil in high permeable zone of large throat (chip 1 and chip 4) is higher than that in the high permeable zone of small throat (chip 2 and chip 3). This is because, in the high permeable area of the large throat, the residual oil droplets formed by fingering make it more difficult to remove oil.

3.1.3. Residual Oil Distribution under Different Wall Wetting Degree

The comprehensive characteristics of rock-fluid in the wettability of soil particles depend on the interfacial tension between rock-fluid and fluid and the adsorption of polar substances on the rock surface. Study on wall wettability plays an important role in reducing displacement residual oil. The simulation shows that under different wall wettability conditions, the residual oil in the model will change. Taking chip 1 as an example, the distribution of residual oil was discussed when the wall contact angles were π/6 rad, π/3 rad, π/2 rad, 2π/3 rad and 5π/6 rad. In the above wall, the contact angle of π/6 rad represents the most hydrophilic of the wall, and the contact angle of 5π/6 rad represents the most lipophilic of the wall. Figure 6 shows the oil volume fraction variation at different time under different wall wettability.

At a wall contact angle of 5pi/6 rad, the water phase displaces the oil phase to form an oil film on the particles in the hyperpermeable zone, and at 0.2 min the water phase breaks through the hyperpermeable zone to form one of the narrowest dominant channels with the smallest ripple area and the largest amount of residual oil. As the wall contact angle decreases, the time required for the water phase to break through the hyperpermeable zone increases and the amount of residual oil decreases. When the wall surface changes from oleophilic to hydrophilic, the oil phase in the low permeable zone begins to transport and the amount of residual oil is further reduced. When the wall contact angle is pi/6, the wall surface is most hydrophilic and the water phase invades from both the hyper- and hypo-permeable zones at the same time, the front end of the water phase repulsion is smoother and the residual oil in the hyper-permeable zone has been completely removed when the proposed steady state is reached, and the overall residual oil is minimal.

The variation of residual oil in the high and low permeable zones with time under different wall wettability of chip 1 is shown in Figure 7. With the continuous increase of wall wettability, the residual oil in the high and low permeable zones of chip 1 decreases. The wettability of the wall is different, and the distribution of oil and water in the throat is also different. Due to the interaction of interfacial tension, the wetting phase is always attached to the wall, and wants to occupy a smaller gap corner and push the non-wetting phase to the middle of the throat. Therefore, after the water flooding of the lipophilic wall through the pore throat, a large number of residual oil is left. With the continuous increase of wall wettability, the oil displacement effect will be increasingly better.

3.1.4. Distribution of Residual Oil at Different Flow Rates

The groundwater flow rate at different locations is different. In order to combine with the actual experiment, in this simulation, taking into account the typical groundwater flow rate, taking chip 1 as an example, the flow rates of 1.13, 2.26, 4.52 and 6.78 m/d are applied to analyze the change of residual oil in the chip under different flow rates. Figure 8 is the oil volume fraction diagram of the model varying with time at different flow rates.

It can be seen from the Figure 9 that at different inlet flow rates, the water phases all break through from the high permeable zone and form a dominant channel, and the dominant channel formation time in the high permeable zone will increase with the decrease of the inlet flow rate, and the residual oil volume in the high permeable zone does not differ much when reaching the proposed steady state; for the low permeable zone, although it is difficult to displace the oil phase as a whole, the residual oil volume will decrease with the decrease of the inlet velocity. It can be seen that in the non-homogeneous zone, the low flow velocity will increase the ripple area in the low permeable zone, which helps to reduce the residual oil volume.

3.2. Effects of Different Flow Rates on Concentration Distribution of Chemotactic Bacteria

Toluene in the oil phase is a chemoattractant of chemotactic bacteria. The diffusion and dissolution of toluene from the oil phase to the water phase determine the distribution of chemotactic bacteria. Taking chip 1 as an example, the concentration distribution of chemotactic bacteria at different flow rates was analyzed. The specific method to find out the concentration distribution of chemotactic bacteria at the oil–water two-phase interface is to describe the contour when the phase field variable φ = 0, and then obtain the concentration of the chemotactic bacteria.

The concentration of the chemotactic bacteria at the interface between oil and water was 1.1~1.5 mol/m³ at the front end of the oil–water interface in the hypotonic zone and 1.05~1.1 mol/m³ at the oil–water interface in the hypertonic zone at an inlet flow rate of 1.13 m/d. The concentration of chemotactic bacteria at the junction of the hypotonic and hypertonic zones was 1.2~1.8 mol/m³ at the front end of the hypotonic zone and 1.1~1.3 mol/m³ at the oil–water interface in the hypertonic zone, at an inlet flow rate of 2.26 m/d. The concentration of chemotactic bacteria at the oil–water interface in the front end was 1.2~1.8 mol/m³, and the concentration of chemotactic bacteria at the oil–water interface in the hypertonic zone was 1.1~1.3 mol/m³. At an inlet flow rate of 4.52 m/d, the concentration of chemotactic bacteria at the oil–water interface in the front end at the junction of the hypotonic zone and the hypertonic zone was 1.3~1.5 mol/m³, and the concentration of chemotactic bacteria at the oil–water interface in the hypertonic zone was 1.2~1.6 mol/m³. At an inlet flow rate of 6.78 m/d, the chemotactic bacteria were mainly enriched at the oil–water interface in the hypertonic zone, with concentrations of up to 1.3~2 mol/m³.

The curve of the concentration of chemotactic bacteria at a certain point at the interface with time is shown below.

It can be seen from Figure 10 that the concentration of chemotactic bacteria changes with time at a certain point at the oil–water two-phase interface at different flow rates. When the water phase carrying chemotactic bacteria flows through this point, the chemotactic bacteria are rapidly enriched. The greater the flow rate is, the earlier the time is to reach the steady state, and the concentration of chemotactic bacteria at the phase interface decreases with the decrease of the flow rate at the steady state. This is because, when the oil phase is displaced, the toluene in the oil dissolves into the water phase to form a concentration transition zone, and the higher the flow rate is, the steeper the concentration gradient is. Therefore, the enrichment degree of chemotactic bacteria is higher at higher flow rates.

One interesting finding is the transient peak before the steady state in each curve. One possible explanation is that, as the toluene diffuses from the oil phase to the water phase at the phase interface under the flushing of water, a high concentration gradient is formed immediately, and the chemotactic bacteria rapidly enrich to a certain concentration, but as the toluene continues to diffuse into the water phase, the concentration gradient decreases, and so the concentration of chemotactic bacteria decreases accordingly, finally the chemotactic bacteria in the water phase are continuously transported to the oil–-water interface, and the concentration of chemotactic bacteria slowly rises until the proposed steady state. Therefore, the concentration of chemotactic bacteria first rises to a certain peak value and then falls, and then continues to rise to a plateau.

By deriving the average concentration data of chemotactic bacteria at each time point under different flow rates, the variation curve of the average concentration of chemotactic bacteria with time is shown in Figure 11. It can be seen from the figure that, when the inlet flow rates were 1.13, 2.26, 4.52 and 6.78 m/d, the time to reach the quasi-steady state decreased in turn, and when the quasi-steady state was reached, the average concentration of chemotactic bacteria at low flow rate was higher than that at high flow rate. This is because chemotactic bacteria are more likely to be washed out of the chip by water at a high inlet flow rate, which is not conducive to the enrichment of chemotactic bacteria in the transition zone from chemotactic bacteria to pollutant concentration. The high enrichment at high flow rates near the oil–water interface is offset by the fact that at high flow rates away from the interface, the bacteria are subject to higher shear forces and are more easily flushed away. Therefore, the total amount of chemotactic bacteria in the system tended to be the same at different flow rates, although the enrichment amount at the oil–water interface was different. Still, in general, chemotactic bacteria will be more retained throughout the porous media system than non-treating common bacteria, and have more enrichment at the source of contamination, theoretically enhancing oil drive and remediation of oil contamination.

4. Conclusions

In this paper, the physical fields of oil–water two-phase flow and oil-phase solute convection and diffusion in water are successfully coupled. Toluene is used as the contaminant dissolved in the oil phase and the chemoattractant of chemotactic bacteria. Based on the control equation of phase field model and the convection–diffusion equation of chemoattractant, the oil–water two-phase transfer field is simulated. Based on the convection–diffusion equation of bacteria and the chemotaxis velocity equation, the migration of chemotactic bacteria in the oil–water two-phase transfer field is simulated and the following conclusions are drawn:

(1)

For the distribution of residual oil volume, under all of the simulation conditions, the residual oil volume in the high permeable area is significantly lower than that in the low permeable area, and residual oil droplets are easy to form in the small throat, high permeable area, and a residual oil belt is formed in the area connected with the low permeable area in the large throat, high permeable area. It is difficult for the water phase to invade the large-size, low-permeable zone, and the corresponding water phase begins to invade the small-size low-permeable zone after 1 min;

(2)

With the increase of the hydrophilic degree of the wetting wall, the overall residual oil is less, and the wall hydrophilicity promotes the oil phase to be displaced from the low permeable area. However, even when the wall wetting angle is π/6 rad, there is still a large quantity of residual oil in the low permeable area;

(3)

The higher the inlet flow rate is, the shorter the time is that is required to form the dominant channel in the high permeable area, and the relationship between the residual oil in the high permeable area and the flow rate is not significant. However, when the water phase drives the oil phase at a low flow rate (1.13 or 2.26 m/d), the water phase in 1 min tends to invade the low permeable area;

(4)

For the density distribution of chemotactic bacteria, from the beginning to the end, the chemotactic bacteria are largely gathered at the oil–water interface, and the number is more than that in the water phase. Therefore, chemotactic bacteria can react to the concentration gradient of the chemoattractant toluene formed by the oil–water two-phase transfer field, and enrich near the pollutants;

(5)

The lower the inlet velocity, the higher the average density of chemotactic bacteria in the chip at quasi-steady state. This is because when the inlet velocity is high, the chemotactic bacteria are washed away by water, which is not conducive to enrichment in the pollutant transition zone.