Adaptively-localized-continuation-Newton; Reservoir Simulation Nonlinear Solvers that Converge All the Time

Abstract

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 in the sense of discrete approximations. This stability comes at the expense of transferring the inherent physical stiffness onto the coupled nonlinear residual equations which need to be solved at each time-step. Current reservoir simulators apply safe-guarded variants of Newton’s method, and often can neither guarantee convergence, nor provide estimates of the relation between convergence rate and time-step size. In practice, time-step chops become necessary, and are guided heuristically. With growing complexity, such as in thermally reactive compositional models, this can lead to substantial losses in computational effort, and prohibitively small time-steps. We establish an alternate class of nonlinear iteration that both converges, and associates a time-step to each iteration. Moreover, the linear solution process within each iteration is performed locally. By casting the nonlinear residual for a given time-step as an initial-value-problem, we formulate a solution process that associates a time-step 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 a standard Newton counterpart whenever it does converge. Finally, by exploiting the local nature of nonlinear waves that is typical to all multiphase problems, we establish a linear solution process that performs computation only where necessary. That is, given a linear convergence tolerance, we identify the 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 large-scale simulation examples are presented, and the results demonstrate the robustness of the proposed method as well as its performance.