Decoupling and Block Preconditioning for Sedimentary Basin Simulations

Abstract

Abstract. The simulation of sedimentary basins aims at reconstructing its historical evolution in order to provide quantitative predictions about phenomena leading to hydrocarbon accumulations. The kernel of this simulation is the numerical solution of a complex system of partial differential equations of mixed parabolic-hyperbolic type. A discretisation and linearisation of this system leads to large ill-conditioned non-symmetric linear systems with three unknowns per mesh element. The preconditioning which we will present for these systems consists in three stages: (i) a local decoupling of the equations which (in addition) aims at concentrating the elliptic part of the system in the "pressure block"; (ii) an efficient preconditioning of the pressure block using AMG; (iii) the "recoupling" of the equations. In all our numerical tests on real case studies we observed a reduction of the CPU-time for the linear solver (up to a factor 4.3 with respect to the current preconditioner ILU(0)) and almost no degradation with respect to physical and numerical parameters.

High Performance Computing Linear Algorithms for Two-Phase Flow in Porous Media

“…In this case, PETSc provides the solution on both pressure and saturation variations at own vertices for each process p ∈ P. The variations at the unknowns at the ghost vertices are then obtained by synchronization. As discussed in the introduction of this paper, this strategy implies the implementation of Combinative preconditioners [11]. The following sequential scheme proposes a related but simpler to implement approach.…”

“…But for more general situations of two-phase flow (such as the case considered in the numerical example of this paper), this strategy fails. This has led to the development of efficient Combinative-AMG preconditioners [11], combining typically an AMG preconditioner on the pressure block with an ILU preconditioner on the full system. This paper focuses on an alternative algorithm, based on the PETSc environment, for the resolution of the linear systems issued from a fully implicit scheme for the approximation of (1-2).…”

Algebraic Domain Decomposition Methods for Highly Heterogeneous Problems

“…Nested factorization [AC83] and CPR-AMG [LVW01], [SMW03] are the main state of the art preconditioners currently used in industrial reservoir simulators. Nevertheless they still suffer from major drawbacks that should be addressed in response to the evolution of the computing architectures, and the demand for more complex physics and geology in reservoir and basin simulations.…”

Parallel Preconditioning for Sedimentary Basin Simulations

“…The idea for our preconditioning strategy is based on the methods developed in [2,3,5,9] for the related problem of oil reservoir simulation and is described in detail in [7]. It consists of three steps.…”

“…pressure, geostatic load, and oil saturation. This article describes the parallel version of a preconditioner for these systems, presented in its sequential form in [7]. It consists of three steps: in the first step a local decoupling of the pressure and saturation unknowns aims at concentrating in the "pressure block" the elliptic part of the system which is then, in the second step, preconditioned by AMG.…”