Adaptively Localized Continuation-Newton Method—Nonlinear Solvers That Converge All the Time

Abstract

Summary Growing interest in understanding, predicting, and controlling advanced oil-recovery methods emphasizes the importance of numerical methods that exploit the nature of the underlying physics. The fully implicit method offers unconditional stability of the discrete approximations. This stability comes at the expense of transferring the inherent physical stiffness onto the coupled nonlinear residual equations that are solved at each timestep. Current reservoir simulators apply safeguarded variants of Newton's method that can neither guarantee convergence nor provide estimates of the relation between convergence rate and timestep size. In practice, timestep chops become necessary and are guided heuristically. With growing complexity, such as in thermally reactive compositional flows, convergence difficulties can lead to substantial losses in computational effort and prohibitively small timesteps. We establish an alternative class of nonlinear iteration that converges and associates a timestep to each iteration. Moreover, the linear solution process within each iteration is performed locally. By casting the nonlinear residual equations for a given timestep as an initial-value problem, we formulate a continuation-based solution process that associates a timestep size with each iteration. Subsequently, no iterations are wasted and a solution is always attainable. Moreover, we show that the rate of progression is as rapid as that for a convergent standard Newton method. Moreover, by exploiting the local nature of nonlinear wave propagation typical to multiphase-flow problems, we establish a linear solution process that performs computation only where necessary. That is, given a linear convergence tolerance, we identify a minimal subset of solution components that will change by more than the specified tolerance. Using this a priori criterion, each linear step solves a reduced system of equations. Several challenging examples are presented, and the results demonstrate the robustness and computational efficiency of the proposed method.