An adaptive time stepping algorithm for IMPES with high order polynomial extrapolation

Abstract

Two-phase flows in oil reservoirs can be modeled by a coupled system of elliptic and hyperbolic partial differential equations. The transport velocity of the multiphase fluid system is related to the pressure through Darcy’s law and it is coupled to a conservation law for the saturation variable of one of the phases. A time step of the classical IMPES (IMplicit Pressure Explicit Saturation) method consists of first solving the elliptic problem for pressure and Darcy velocity, and then updating the saturation with an explicit numerical scheme for conservation laws. This method is very computationally costly, since the time-consuming elliptic solver must be invoked at time intervals defined by the stability limit of the hyperbolic solver. A popular variant is not to update the velocity at all hyperbolic time steps, but to skip a fixed number C of them, with C determined by the user. In this work we propose a more accurate and systematic procedure for time stepping in IMPES codes. The velocity is updated at all transport time steps, though the elliptic solver is only invoked every C steps. In the time steps at which the elliptic problem is not solved, the velocity is extrapolated from previously computed values with polynomials of high degree. Further, we introduce an error estimator that allows for the number C to be adaptively determined without user intervention. The algorithm was tested in several relevant benchmark problems. This allowed for the optimization of its parameters and comparisons with previous variants. The results show that the proposed algorithm is very stable, reliable and time-cost effective. It is also easily implemented in pre-existent IMPES codes.