Potential-based reduced Newton algorithm for nonlinear multiphase flow in porous media

We present a phase-based potential ordering that is an extension of the Cascade ordering introduced by Appleyard and Cheshire [John R. Appleyard, Ian M. Cheshire, The cascade method for accelerated convergence in implicit simulators, in: European Petroleum Conference, 1982, pp. 113–122]. The proposed ordering is valid for both two-phase and three-phase flow, and it can handle countercurrent flow due to gravity and/or capillarity. We show how this ordering can be used to reduce the nonlinear algebraic system that arises from the fully-implicit method (FIM) into one with only pressure dependence. The potential-based reduced Newton algorithm is then obtained by applying Newton’s method to this reduced-order system. Numerical evidence shows that our potential-based reduced Newton solver is able to converge for time steps that are much larger than what the standard Newton’s method can handle. In addition, whenever standard Newton converges, so does the reduced Newton algorithm, and the number of global nonlinear iterations required for convergence is significantly reduced compared with the standard Newton’s method.