The finite-control volume technique is used to solve the convection-dispersion equation in radial coordinates. In the dispersion model presented, the dispersion coefficient is dependent on both velocity and diffusion coefficient. The finite-control volume scheme is shown to have negligible numerical dispersion. The code is verified by performing a convergence study to show that the numerical results are independent of the mesh and time-step sizes.
The solution allows analyzing the dispersion transport of a slug in a steady radial flow from an injection well fully penetrating a homogeneous reservoir of uniform thickness and finite arial extent. Results show that the dispersive mixing zone does not necessarily grow in proportion with the square root of time for all times, unlike for the linear displacement. Results reveal a typical minimum of the dispersive mixing zone with time for radial displacement. It appears that at some radial distance close to the wellbore, the dimensionless mixing zone decreases with the square root of time since there is an accumulation of the miscible slug in this particular space domain. However, when the dissolution of the slug starts taking effect particularly at large distances away from the injection point, the mixing zone begins to increase with increasing square root of time. These results indicate that the optimal slug size of miscible displacement is time-dependent.
At a fixed time value, the higher the value of dispersivity, the more mixing takes place, and hence the greater the concentration of the miscible fluid becomes. A higher dispersivity value aggravates the dissolution of the slug at the toe of the concentration profile, but gives rise to a larger concentration at the tail. The smaller the porosity value is, the larger the rate of change of concentration. This is expected since a smaller value of porosity acts like a choke on the porous medium and results on a greater degree of convective mixing than diffusion mixing.
dispersion, porous media, finite-volume method
