The geometry chosen for the development of the method is a two-dimensional box in which there is one or several wells.

The wellbore influence is simply taken into account by adding a source term in the mass conservation equations. In order to have a flexibility in the positions of the wells, a constant, associated which this source term, is defined in the whole domain and must be equal to one where a wellbore is present or zero otherwise.

The source term must be negative as in the well the oil is extracted. We took a very simple model of well in order to implement the first simulations. For all the simulations in this project the source term will be equal to the local flow rate.

On the Top of the domain we defined a condition of constant pressure (Pref equal to the atmospheric pressure for example).On the rest of domain boundaries we have walls. Then we put a condition of zero velocity on those parts. When we take into account the gravity, the boundary condition on the pressure gradient on the bottom of the domain will be calculated as a function of the gravity field.