The effective use of streamline simulators for flow simulation of multi-million cell detailed 3D models relies on the ability to take large simulation time-steps with few pressure solutions. For processes that are convective, the streamline approach works quite well while for flow simulation with compressibility, strong capillarity or strong gravity terms, the maximum time step size may be substantially reduced, limiting the utility of streamline simulation. This is the case when applying the conventional streamline operator-splitting approach, where the nonlinear terms associated with capillarity and gravity limit the time step. Earlier studies have shown the importance of an "anti-diffusive" correction in which numerical dispersion from the convective solution must be removed before capillary pressure can be accurately modeled. Evaluation of the anti-diffusive term involves the solution of a local Riemann problem which, unfortunately, is difficult to determine in full field multi-dimensional problems with heterogeneity, and spatially variable viscosity, fluid velocity, and saturations. The alternative approach is to perform the operating splitting calculation with very small time-steps to minimize the numerical dispersion, but this is not an effective simulation strategy.
In our approach, the equations are reformulated so that all of the flux terms including capillarity and gravity forces are solved simultaneously with the streamline convection equations. We utilize an orthogonal projection method in which the fluxes of capillary and gravity are separated into components parallel and orthogonal to the total velocity. Fluxes parallel to total velocity are included within the solution of the convective flow equations on streamlines. The remaining terms are calculated on the underlying three dimensional grid. With the introduction of this orthogonal-projection, there is no longer a need for an anti-diffusive correction. This formulation still uses an operator splitting approach, but the size of the remaining transverse flux correction terms are reduced, allowing for large time steps.
We demonstrate the utility and validity of our approach using a series of increasingly complex numerical experiments in 1D and 2D including the 3D SPE10 reservoir model. We compare our results to a commercial finite difference simulator, and to a streamline simulator using a conventional operator splitting approach, but without the anti-diffusive correction. We obtain a good match to the saturation distribution and production profile using large time steps, compared to the small time steps required for conventional operator splitting. The 2D and 3D examples clearly demonstrate the effectiveness of the orthogonal projection approach in minimizing the transverse flux allowing for the larger time steps. It also provides a systematic means of including additional displacement mechanisms in the future.