Preconditioning for Efficiently Applying Algebraic Multigrid in Fully Implicit Reservoir Simulations

Abstract

Fully implicit black-oil simulations result in huge, often very-illconditioned, linear systems of equations for different unknowns (e.g., pressure and saturations). It is well-known that the underlying Jacobian matrices contain both hyperbolic and nearly elliptic subsystems (corresponding to saturations and pressure, respectively). Because a reservoir simulation is typically driven by the behavior of the pressure, constrained-pressure-residual (CPR)type two-stage preconditioning methods to solve the coupled linear systems are a natural choice and still belong to the most popular approaches. After a suitable extraction and decoupling, the computationally most costly step in such two-stage methods consists in solving the elliptic subsystems accurately enough. Algebraic multigrid (AMG) provides a technique to solve elliptic linear equations very efficiently. Hence, in recent years, corresponding CPR-AMG approaches have been extensively used in practice.Unfortunately, if applied in a straightforward manner, CPR-AMG does not always work as expected. In this paper, we discuss the reasons for the lack of robustness observed in practice, and present remedies. More precisely, we will propose a preconditioning strategy (based on a suitable combination of left and right preconditioning of the Jacobian matrix) that aims at a compromise between the solvability of the pressure subproblem by AMG and the needs of the outer CPR process. The robustness of this new preconditioning strategy will be demonstrated for several industrial test cases, some of which are very ill-conditioned. Furthermore, we will demonstrate that CPR-AMG can be interpreted in a natural way as a special AMG process applied directly to the coupled Jacobian systems.

Accelerating multiscale simulation of complex geomodels by use of dynamically adapted basis functions

“…Popular decoupling strategies include alternate-block factorization (ABF) [2], IMPES and quasi-IMPES (IMplicit Pressure, Explicit Saturation) [6,21], and dynamic row sum (DRS) [10]. Herein, we follow the IM-PES and quasi-IMPES approaches.…”

Robust Fully-Implicit Coupled Multiphase-Flow and Geomechanics Simulation

“…A number of proven iterative linear solvers are used by the coupled simulator, including GMRES (default for flow), PCG, BiCGSTAB (default for geomechanics) (Saad 2003), and AMG (Briggs et al 2000). A number of preconditioning approaches are also available, including block Jacobi, ILU(n), SCMG (Alpak and Wheeler 2012), and CPR-AMG (only for flow) (Gries et al 2014). …”

“…A classical 1D consolidation problem is considered as a simplified coupled flow and geomechanical deformation test case that involves both porous-medium compaction and fluid-pressure diffusion (Gu et al 2011). Fig.…”

“…Case 4 is a test problem first described by Gutierrez and Lewis (1998) and later adopted by Gutierrez et al (2001), Dean et al (2006), Samier et al (2006), Gu et al (2011), and Alpak and Wheeler (2012). It models the depletion process in a soft hydrocarbon reservoir encased within a stiff nonpay region (Fig.…”

Efficient reordered nonlinear Gauss–Seidel solvers with higher order for black-oil models

“…The pressure equation is usually discretized by a first-order, two-point flux-approximation scheme, giving a discrete problem that can be efficiently solved by a multigrid [33,11] or multiscale method [22]. For the transport equations, a number of methods aim to arXiv:2001.01630v1 [math.NA] 6 Jan 2020 accelerate computations by utilizing inherent locality and co-current flow properties of hyperbolic equations.…”