Abstract

In reservoir simulation, numerical solutions to partial differential equations are required. The system of nonlinear PDEs is linearized utilizing Newton-Raphson iteration. Successive solution to the linear system is required. Direct solutions are slow and sale poorly. The iterative methods or solvers are commonly adopted for reservoir simulation, but still may consume 75% to 90% of simulation time.

The fast and robust solvers (Krylov Solvers) include ORTHOMIN (orthogonal minimum residual), GMRES (generalized minimum residual) and BiCGStab (biconjugate gradient stabilized). The good strategy should focus on a good pre-conditioner rather than the best accelerator, and best pre-conditioners are often problem dependent. Properties of a good preconditioner should be scalable, parallizable, robust over large problem class and so on.

Reordering can reduce fill-in, introduce parallelism and improve stability. Multi-stage pre-conditioners can decompose system into subsystems, and sensitive to mixed character PDE’s. The constrained pressure residual weakens coupling or decouples pressure variables from others. The preconditioning can be guided by recapturing physics such as exploiting physical connectivity to define pre-conditioners, percolation technique and searching transmissibility or permeability.