Numerical issues concerning the simulation of foam EOR

Numerical issues concerning the simulation of foam EOR

Petroleum Engineering, Delft University of Technology,

The Netherlands February 26, 2013

If secondary hydrocarbon recovery methods, like water flooding, fail because of the occurrence of viscous fingering one can turn to an enhanced oil recovery method (EOR) like the injection of foam. The generation of foam can be described by a set of partial differential equations with strongly nonlinear functions, which impose challenges for the numerical modeling. Former studies by Zanganeh [2011] and Ashoori [2012] show the occurrence of strongly temporally oscillating solutions when using forward simulation models, that are entirely due to discretization artifacts.

To analyze these problems, we study the dynamics of a simple foam model based on the Buckley-Leverett equation. Whereas the Buckley-Leverett flux is a smooth function of water saturation, the foam will cause a rapid increase of the flux function over a very small saturation scale. Consequently the derivatives of the flux function can become extremely large and impose a severe constraint on the time step due to the CFL condition. The solution of the exact Riemann problem for the simple foam model forms the basis of a tailor made approximate Riemann solver [Toro, 2009] that captures all the characteristic waves accurately and mollifies the constraint on the time step. At the same time we would like to use higher-order schemes, to avoid unphysical oscillations near the foam front. In order to do this we make use of several total variation diminishing (TVD) and oscillation diminishing finite volume schemes that preserve the stability of the solutions and are adapted to our specific problem constraints.

Until now, the methods applied to foam EOR processes are only first-order accurate and do not incorpo-rate stabilization near the foam front as far as we know. We expect that for the 1D model the proposed methods will give stable results, which we can verify by comparing with the analytical solutions. More complications may arise for the 2D case when gravitational and capillary pressure forces play a significant role.

References

E. Ashoori. Foam for Enhanced Oil Recovery: Modelling and Analytical Solutions. PhD thesis, Delft University of Tecnology, 2012.

E.F. Toro. Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer-Verlag, 2009.

M.N. Zanganeh. Simulation and Optimization of Foam EOR processes. PhD thesis, Delft University of Tecnology, 2011.