A posteriori error estimates, stopping criteria, and adaptivity for multiphase compositional Darcy flows in porous media

Abstract

International audienceIn this paper we derive a posteriori error estimates for the compositional model of multiphase Darcy flow in porous media, consisting of a system of strongly coupled nonlinear unsteady partial differential and algebraic equations. We show how to control the dual norm of the residual augmented by a nonconformity evaluation term by fully computable estimators. We then decompose the estimators into the space, time, linearization, and algebraic error components. This allows to formulate criteria for stopping the iterative algebraic solver and the iterative linearization solver when the corresponding error components do not affect significantly the overall error. Moreover, the spatial and temporal error components can be balanced by time step and space mesh adaptation. Our analysis applies to a broad class of standard numerical methods, and is independent of the linearization and of the iterative algebraic solvers employed. We exemplify it for the two-point finite volume method with fully implicit Euler time stepping, the Newton linearization, and the GMRes algebraic solver. Numerical results on two real-life reservoir engineering examples confirm that significant computational gains can be achieved thanks to our adaptive stopping criteria, already on fixed meshes, without any noticeable loss of precision

A Review of Recent Advances in Discretization Methods,a PosterioriError Analysis, and Adaptive Algorithms for Numerical Modeling in Geosciences

“…Thus the overall error can be controlled and moreover its individual components identified. Similarly to Section 3.4.2, entirely adaptive algorithms were proposed for the compositional multi-phase flow case [113]. Results similar to those reported in Figures 13 and 14 were obtained, enabling in particular substantial computational gains just by employing adaptive stopping criteria for linear and nonlinear solvers.…”

“…We recently in [113] adapted the developments presented in Section 3.4 for two-phase flows to this setting.…”

“…In [113], we have in particular derived an equivalent of Theorem 3 for the compositional multi-phase flow. Thus the overall error can be controlled and moreover its individual components identified.…”

Reservoir Simulator Runtime Enhancement Based ona PosterioriError Estimation Techniques

“…First rigorous approachs can be found in [2][3][4] suitable for unsteady nonlinear models, and [5][6][7] for degenerated problems. Recently, abstract frameworks for a posteriori estimates appeared, for two-phase flow [8,9], multiphase compositional model [10], and the thermal multiphase compositional global model [11]. In these frameworks the dual norm of the residual augmented by a nonconformity evaluation term is controlled by fully computable estimators decomposed into space, time, linearization, and algebraic error components.…”

“…In this paper we propose an engineering approach of the a posteriori error estimates, provided by recent works [10,11], in order to establish an efficient computation of the estimators in such a way that we can ensure important computational savings for realistic complex models. Our approach is based on providing a low-cost evaluation formula by avoiding flux reconstructions.…”