Localized Linear Systems in Sequential Implicit Simulation of Two-Phase Flow and Transport

Abstract

Summary Implicit-reservoir-simulation models offer improved robustness compared with semi-implicit or explicit alternatives. The implicit treatment gives rise to a large nonlinear algebraic system of equations that must be solved at each timestep. Newton-like iterative methods are often used to solve these nonlinear systems. At each nonlinear iteration, large and sparse linear systems must be solved to obtain the Newton update vector. It is observed that these computed Newton updates are often sparse, even though the sum of the Newton updates over a converged iteration may not be. Sparsity in the Newton update suggests the presence of a spatially localized propagation of corrections along the nonlinear iteration sequence. Substantial computational savings may be realized by restricting the linear-solution process to obtain only the nonzero update elements. This requires an a priori identification of the set of nonzero update elements. To preserve the convergence behavior of the original Newton-like process, it is necessary to avoid missing any nonzero element in the identification procedure. This ensures that the localized and full linear computations result in the same solution. As a first step toward the development of such a localization method for general fully implicit simulation, the focus is on sequential implicit methods for general two-phase flow. Theoretically conservative, a priori estimates of the anticipated Newton-update sparsity pattern are derived. The key to the derivation of these estimates is in forming and solving simplified forms of infinite-dimensional Newton iteration for the semidiscrete residual equations. Upon projection onto the discrete mesh, the analytical estimates produce a conservative indication on the update's sparsity pattern. The algorithm is applied to several large-scale computational examples, and more than a 10-fold reduction in simulation time is attained. The results of the localized and full simulations are identical, as is the nonlinear convergence behavior.