An Adaptive Time Marching Strategy for IMPES

Abstract

In a multiphase fluid system, the transport velocity can be 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. The resulting coupled system of elliptic and hyperbolic partial differential equations is used to the modeling of, for example, two-phase flows in oil reservoirs. The classical IMPES (IMplicit Pressure Explicit Saturation) method first solves the elliptic problem for the pressure and flux, and then updates the saturation with an explicit hyperbolic solver. This method is very costly, since the expensive elliptic solver must be invoked at time intervals defined by the stability limit of the hyperbolic solver. It is popular among users to update the flux every C time steps, keeping it frozen in between, with C determined by the user. In this work we propose a more accurate handling of the velocity and an automatic procedure for the selection of C in IMPES codes. In the time steps at which the elliptic problem is not solved, the flux is extrapolated from previously computed values with polynomials of high degree. We also introduce an error estimator from which the correct value of C can be derived without user intervention. The algorithm is very easy to implement. The results show that the proposed algorithm is stable, reliable and cost effective.