## 2 Materials and Methods
### 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 [PERSON] 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 [PERSON] et al. [21]. The two equations were solved and the density distribution of bacteria was obtained. [PERSON] 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:  
\[\frac{\partial b}{\partial t}=D_{e,b}\frac{\partial^{2}b}{\partial x^{2}}+D_{e,b}\frac{\partial^{2}b}{\partial y^{2}}-\frac{v_{x}\partial b}{\partial x}- \frac{v_{y}\partial b}{\partial y}-\frac{v_{Chx}\partial b}{\partial x}-\frac{ v_{Chy}\partial b}{\partial y}, \tag{14}\]  
\(b\)--Bacterial density, mol/m\({}^{3}\);  
\(t\)--Time, s;  
\(D_{e,b}\)--Effective random movement coefficient of bacteria, m\({}^{2}\)/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:  
\[D_{e,b}=\varepsilon_{p}D_{b}, \tag{15}\]  
\(D_{b}\)--Random motion coefficient of bacteria, m\({}^{2}\)/s;  
\(\varepsilon_{p}\)--Calculation porosity of pore network, dimensionless.  
In the two-dimensional plane, the bacterial chemotaxis rate equation is:  
\[\left\{\begin{array}{l}v_{Chx}=\frac{2}{3}v_{b}{\rm tanh}\left(\frac{x_{0}} {2v_{b}}\frac{K_{c}}{(K_{c}+a)^{2}}\frac{\partial a}{\partial x}\right)\\ v_{Chy}=\frac{2}{3}v_{b}{\rm tanh}\left(\frac{x_{0}}{2v_{b}}\frac{K_{c}}{(K_{c} +a)^{2}}\frac{\partial a}{\partial y}\right)\end{array}\right., \tag{16}\]  
\(v_{b}\)--Bacterial swimming speed, m/s;  
\(\chi_{0}\)--The chemotactic sensitivity coefficient, m\({}^{2}\)/s;  
\(K_{c}\)--Chemokine receptor constant, mol/m\({}^{3}\);  
\(a\)--Chemoattractant concentration, mol/m\({}^{3}\).  
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.