This work develops the theoretical basis for a promising safeguarding strategy that is applicable to improving the robustness and efficiency of Newton-like methods for general physics implicit reservoir simulation. The method is demonstrated for a range of models with growing complexity, including thermal compositional problem, and is compared to state-of-the-art alternatives.
The damping algorithm is based on viewing the Newton iteration as a forward Euler discretization of the Newton flow equations. Three alternative a posteriori local discretization error estimates are developed. The first and second is are based on computable estimates of the norm of the derivative of the Newton step with respect to step-length. This estimate provides an accurate measure of the departure from the Newton flow path. The third estimate is based inspired by Richardson extrapolation idea. We propose to control the Newton step length by limiting the estimated local discretization error. The control strategy is conservative far from the solution, and can be shown to result in the standard Newton method otherwise.
Computational results are presented for a series of simulation problems with increasing complexity. First, results for two phase flow simulations demonstrate that the proposed method is competitive with, but not superior to current state-of-the-art strategies. For thermal, reactive, and multicomponent problems the method is compared to line-search methods. Superior robustness and computational efficiency are observed, and the iteration converges for significantly larger time-step sizes. These comparisons show that absent of tailored ad hoc strategies such as those recently proposed for black oil simulation, the proposed strategy is more robust and efficient than the current state-of-the art. This is demonstrated by testing the required number of iterations to convergence for different time-step sizes as well as during the course of full simulations.
While several novel Newton damping strategies and alternative discretization with improved differentiability have been proposed and successfully demonstrated for multiphase flow simulation, this work provides a first step towards tailored robust safeguards for complex physics.