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 guide 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 multi-phase 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 examples are presented, and the results demonstrate the robustness of the proposed method as well as its performance.
Introduction Multi phase, multi component, flows through subsurface porous media couple several physical phenomena with vastly differing characteristic scales. The fastest processes such as component phase equilibria occur instantaneously, and are modeled as nonlinear algebraic constraints. Mass conservation laws govern the transport of species propagating through the flow field. These transport phenomena are near-hyperbolic, and they evolve with a finite domain of dependence. Moreover, the flow field itself is transient, and evolves with parabolic or elliptic character. The underlying constitutive relations, such as that for the velocity of a phase, couple the variables across governing equations in a strongly nonlinear manner. Consequently, one challenge in modeling large scale flows through complex media is to honor such coupling without sacrificing stability. Additional sources of complexity include the heterogeneity of the underlying porous media, body forces, and the presence of wells.