Adaptive algebraic multiscale solver for compressible flow in heterogeneous porous media

Abstract

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.

Multiscale formulation for coupled flow-heat equations arising from single-phase flow in fractured geothermal reservoirs

“…Equation 20 is formulated based on an equivalent incompressible system equation. This formulation is proven [34] to be the most efficient strategy (based on CPU measurements) because it eliminates the need to frequently update the local basis function, while the fully compressible coarse-scale system takes care of the global compressibility (and time-dependent) effects.…”

“…x x x ν+1/2 = P P Px x x ν+1/2 c = P P P(R R RA A A ν P P P) −1 R R Rf f f ν , (34) or in residual form, δx δx δx ν+1/2 = P P Pδ δ δx x x ν+1/2 c = P P P(R R RA A A ν P P P) −1 R R Rr r r ν ,…”

Improving the Computational Efficiency of Approximate Gradients Using a Multiscale Reservoir Simulation Framework

“…This accounts for all computations performed by the i-MSFV solution strategy, i.e solution of the coarse system plus fine-scale smoothing. Note that, although the construction of basis function adds an overhead to the total simulation time (Tene et al 2015), we highlight that in our experiments basis functions are only built once. Also, it is well-known that the linear system solution is the most time-consuming part in reservoir simulation framework.…”

“…One of the ways to improve the computational efficiency of forward simulation models in optimization is the use of Reduced Order Models (ROM) (Cardoso et al 2010;Jansen and Durlofsky 2016;van Doren et al 2006). Alternatively, there, has been an increase in the applicability of different simulation strategies to speed up the computational process.…”

“…MS methods aim to solve the equations at a coarser scale, yet preserving the fine scale description. MS methods have increasingly been demonstrated as an efficient and accurate technique for reservoir simulation (Cusini et al 2015;Kozlova et al 2015Kozlova et al , 2016Tene et al 2015). For an overview of important developments in MS we refer to Lie et al (2016).…”