Two-Level Domain Decomposition Methods for Highly Heterogeneous Darcy Equations. Connections with Multiscale Methods

Dolean, Victorita; Jolivet, Pierre; Nataf, Frédéric; Spillane, Nicole; Xiang, Hua

doi:10.2516/ogst/2013206

Cited by 7 publications

(2 citation statements)

References 26 publications

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“…Moreover, due to its conservative property, C-AMS requires only a few iterations per time step to obtain a good quality approximation of the pressure solution for practical purposes. Systematic error estimate analyses for 3D multiphase simulations are a subject of ongoing research and, in addition, the C-AMS performance can be further extended by enrichment of the multiscale operators [29,32,20], and enriched coarse grid geometries on the basis of the underlying fine-scale transmissibility. Both are subjects of our future studies.…”

Section: Discussionmentioning

confidence: 99%

Adaptive algebraic multiscale solver for compressible flow in heterogeneous porous media

Ţene

Wang

Hajibeygi

2015

Journal of Computational Physics

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This paper presents the development of an Adaptive Algebraic Multiscale Solver for Compressible flow (C-AMS) in heterogeneous porous media. Similar to the recently developed AMS for incompressible (linear) flows [Wang et al., JCP, 2014], C-AMS operates by defining primal and dual-coarse blocks on top of the fine-scale grid. These coarse grids facilitate the construction of a conservative (finite volume) coarsescale system and the computation of local basis functions, respectively. However, unlike the incompressible (elliptic) case, the choice of equations to solve for basis functions in compressible problems is not trivial. Therefore, several basis function formulations (incompressible and compressible, with and without accumulation) are considered in order to construct an efficient multiscale prolongation operator. As for the restriction operator, C-AMS allows for both multiscale finite volume (MSFV) and finite element (MSFE) methods. Finally, in order to resolve highfrequency errors, fine-scale (pre-and post-) smoother stages are employed. In order to reduce computational expense, the C-AMS operators (prolongation, restriction, and smoothers) are updated adaptively. In addition to this, the linear system in the Newton-Raphson loop is infrequently updated. Systematic numerical experiments are performed to determine the effect of the various options, outlined above, on the C-AMS convergence behaviour. An efficient C-AMS strategy for heterogeneous 3D compressible problems is developed based on overall CPU times. Finally, C-AMS is compared against an industrial-grade Algebraic MultiGrid (AMG) solver. Results of this comparison illustrate that the C-AMS is quite efficient as a nonlinear solver, even when iterated to machine accuracy.Key words: multiscale methods, compressible flows, heterogeneous porous media, scalable linear solvers, multiscale finite volume method, multiscale finite element method, iterative multiscale methods, algebraic multiscale methods.

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Section: Discussionmentioning

confidence: 99%

Adaptive algebraic multiscale solver for compressible flow in heterogeneous porous media

Ţene

Wang

Hajibeygi

2015

Journal of Computational Physics

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“…Efendiev et al (2011) suggested using eigenvectors of local spectral problems to systematically enrich the initial multiscale space of the MsFE method. Dolean et al (2014) discuss connections between multiscale and domain-decomposition methods and use a Dirichlet-to-Neumann (DtN) map to obtain an eigenvalue problem that gives a more compact set of eigenvectors. DtN maps were also used to formulate an accurate upscaling method, which in turn inspired an enriched version of the MsMFE method (Lie et al 2014).…”

Section: Approximation Properties and Enrichment Strategiesmentioning

confidence: 99%

Successful Application of Multiscale Methods in a Real Reservoir Simulator Environment

Lie¹,

Møyner²,

Natvig

et al. 2016

Proceedings

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For the past ten years or so, a number of so-called multiscale methods have been developed as an alternative approach to upscaling and to accelerate reservoir simulation. The key idea of all these methods is to construct a set of prolongation operators that map between unknowns associated with cells in a fine grid holding the petrophysical properties of the geological reservoir model and unknowns on a coarser grid used for dynamic simulation. The prolongation operators are computed numerically by solving localized flow problems, much in the same way as for flow-based upscaling methods, and can be used to construct a reduced coarse-scale system of flow equations that describe the macro-scale displacement driven by global forces. Unlike effective parameters, the multiscale basis functions have subscale resolution, which ensures that fine-scale heterogeneity is correctly accounted for in a systematic manner. Among all multiscale formulations discussed in the literature, the multiscale restriction-smoothed basis (MsRSB) method has proved to be particularly promising. This method has been implemented in a commercially available simulator and has three main advantages. First, the input grid and its coarse partition can have general polyhedral geometry and unstructured topology. Secondly, MsRSB is accurate and robust when used as an approximate solver and converges relatively fast when used as an iterative fine-scale solver. Finally, the method is formulated on top of a cell-centered, conservative, finitevolume method and is applicable to any flow model for which one can isolate a pressure equation. We discuss numerical challenges posed by contemporary geomodels and report a number of validation cases showing that the MsRSB method is an efficient, robust, and versatile method for simulating complex models of real reservoirs.

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Successful application of multiscale methods in a real reservoir simulator environment

et al. 2017

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Two-Level Domain Decomposition Methods for Highly Heterogeneous Darcy Equations. Connections with Multiscale Methods

Cited by 7 publications

References 26 publications

Adaptive algebraic multiscale solver for compressible flow in heterogeneous porous media

Adaptive algebraic multiscale solver for compressible flow in heterogeneous porous media

Successful Application of Multiscale Methods in a Real Reservoir Simulator Environment

Successful application of multiscale methods in a real reservoir simulator environment

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