Online Adaptive Local-Global Model Reduction for Flows in Heterogeneous Porous Media

2020

J Sci Comput

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In this paper, we consider an online basis enrichment mixed generalized multiscale method with oversampling, for solving flow problems in highly heterogeneous porous media. This is an extension of the online mixed generalized multiscale method [6]. The multiscale online basis functions are computed by solving a Neumann problem in an over-sampled domain, instead of a standard neighborhood of a coarse face. We are motivated by the restricted domain decomposition method.Extensive numerical experiments are presented to demonstrate the performance of our methods for both steady-state flow, and two-phase flow and transport problems.

Section: A Two Phase Flow and Transport Problemmentioning

confidence: 99%

Online Mixed Multiscale Finite Element Method with Oversampling and Its Applications

2020

J Sci Comput

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“…In particular, we consider two-phase flow in a reservoir domain (denoted by Ω). First, we summarize the underlying partial differential equations [9,15]. The basic equation describing the filtration of a fluid through a porous media is the continuity equation, which states that mass is conserved (assuming that the rock and fluids are incompressible):…”

Section: A Two Phase Flow and Transport Problemmentioning

confidence: 99%

A two-grid preconditioner with an adaptive coarse space for flow simulations in highly heterogeneous media

Journal of Computational Physics

2019

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In this paper, we consider flow simulation in highly heterogeneous media that has many practical applications in industry. To enhance mass conservation, we write the elliptic problem in a mixed formulation and introduce a robust two-grid preconditioner to seek the solution. We first need to transform the indefinite saddle problem to a positive definite problem by preprocessing steps. The preconditioner consists of a local smoother and a coarse preconditioner. For the coarse preconditioner, we design an adaptive spectral coarse space motivated by the GMsFEM (Generalized Multiscale Finite Element Method). We test our preconditioner for both Darcy flow and two phase flow and transport simulation in highly heterogeneous porous media. Numerical results show that the proposed preconditioner is highly robust and efficient.

“…In this section, we use our method to solve a two phase flow and transport model problem. First, we summarize the underlying partial differential equations [15,18] to simulate porous media flows. In particular, we consider two-phase flow in a reservoir domain (denoted by Ω) with the assumption that the fluid displacement is driven by viscous effects, that is, we neglect compressibility and gravity for simplicity in our simulations.…”

Section: A Two Phase Flow and Transport Problemmentioning

confidence: 99%

Residual driven online mortar mixed finite element methods and applications

Journal of Computational and Applied Mathematics

2018

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In this paper, we develop an online basis enrichment method with the mortar mixed finite element method, using the oversampling technique, to solve for flow problems in highly heterogeneous media. We first compute a coarse grid solution with a certain number of offline basis functions per edge, which are chosen as standard polynomials basis functions. We then iteratively enrich the multiscale solution space with online multiscale basis functions computed by using residuals. The iterative solution converges to the fine scale solution rapidly. We also propose an oversampling online method to achieve faster convergence speed. The oversampling refers to using larger local regions in computing the online multiscale basis functions. We present extensive numerical experiments(including both 2D and 3D) to demonstrate the performance of our methods for both steady state flow, and two-phase flow and transport problems. In particular, for the time dependent two-phase flow and transport problems, we apply the online method to the initial model, without updating basis along the time evolution. Our numerical results demonstrate that by using a few number of online basis functions, one can achieve a fast convergence.