Real parameters:

$$\rho_{water} /\rho_{oil}$$ (kg/m^{3}) |
$$\nu_{water} /\nu_{oil}$$ (Pa.s) | $$ \sigma_{oil/water} $$ (N/m) |

1000; 800 | 0.001;0.01 | 0.025 |

It is observed a lot of numerical diffusion on the interface with the real parameters, therefore it is difficult to distinguish the two phases,. A large area of the simulation would have the volume fraction of oil around 0.5, which is impossible because oil and water are not miscible.

This phenomenon can be explained by the fact that in a same cell small portion of oil and water are present, therefore the oil volume fraction can not be either zero or one.

To avoid this problem, several modifications can be done, including :

- refine the mesh or increase the time step, but this would increase the time of calculation

- increase the surface tension to avoid the emulsion of the fluid

-increase the viscosity to avoid the effect of the separation of the fluid in others smaller structures.

Firstly, the viscosity and the surface tension are increased in order to analyze if the results are the same as the one with the real parameter with the help of another software fluent.

The parameters of the simulation are:

$$\rho_{water} /\rho_{oil}$$ (kg/m^{3}) |
$$\nu_{water} /\nu_{oil}$$ (Pa.s) | $$ \sigma_{oil/water} $$ (N/m) | $$ dt_{min}/dt_{max}$$ (s) | inlet/outlet lenght (m) | inlet velocity (m/s) | mesh size (m) | mesh size (cells) | inlet Re |

1000; 800 | 1;10 | 0.487 | 1e-7; 1e-5 | 0.056; 0.025 | 0.2 | 0.76x0.44 | 122x88 | 10 |

This simulation shows no numerical diffusion, with the post treatment, the outlet flow rate and the oil remaining and recovery ratio will be analyzed versus time.

For the outlet flow rate:

$$ Q_{outlet}=\frac{1}{n}\sum \limits_{i=1}^n \tau_i v_i L_{outlet}$$

$$ratio_{oil-remaining}=\frac{oil_{initial}-Q_{outlet}t}{oil_{initial}}$$

$$ratio_{oil-recovery}=1-ratio_{oil-remaining}$$

with n: number of outlet cells

$\tau_i$ : oil volume fraction of cell i

v_{i}: outlet velocity of cell i

$$oil_{initial}=\sum \limits_{i=1}^m \sum \limits_{j=1}^l \tau_{ij} S_{cell}$$

m x l: numbers of cells of the domain

$S_{cell}$: area of a cell

To plot the data, the dimensionless number is applied, the flow rate reference is $Q_{inlet}$ and the time reference is $t=\frac{S_{domain}-S_{obstacles}} {L_{inlet} V_{inlet}}$, which is the time for water to fill the domain.

We have flow rate conservation due to the incompressibility of the fluids, therefore when only oil flow out the inlet and outlet flow are equal, after the dimensionless time around 0.3, water begins to flow out, so the oil outflow rate decreases and a change of the gradient in the oil remaining ratio plot which increases because less oil flows out in time.