Online Coupled Generalized Multiscale Finite Element Method for the Poroelasticity Problem in Fractured and Heterogeneous Media
In this paper, we consider the poroelasticity problem in fractured and heterogeneous media. The mathematical model contains a coupled system of equations for fluid pressures and displacements in heterogeneous media. Due to scale disparity, many approaches have been developed for solving detailed fin...
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doaj-c03afb113ade46beabcd58b541dd6a652021-08-26T13:44:43ZengMDPI AGFluids2311-55212021-08-01629829810.3390/fluids6080298Online Coupled Generalized Multiscale Finite Element Method for the Poroelasticity Problem in Fractured and Heterogeneous MediaAleksei Tyrylgin0Maria Vasilyeva1Dmitry Ammosov2Eric T. Chung3Yalchin Efendiev4Multiscale Model Reduction Laboratory, North-Eastern Federal University, 677000 Yakutsk, RussiaDepartment of Mathematics and Statistics, Texas A&M University, Corpus Christi, TX 78412, USAMultiscale Model Reduction Laboratory, North-Eastern Federal University, 677000 Yakutsk, RussiaDepartment of Mathematics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong 999077, ChinaDepartment of Mathematics, Institute for Scientific Computation (ISC), Texas A&M University, College Station, TX 77843, USAIn this paper, we consider the poroelasticity problem in fractured and heterogeneous media. The mathematical model contains a coupled system of equations for fluid pressures and displacements in heterogeneous media. Due to scale disparity, many approaches have been developed for solving detailed fine-grid problems on a coarse grid. However, some approaches can lack good accuracy on a coarse grid and some corrections for coarse-grid solutions are needed. In this paper, we present a coarse-grid approximation based on the generalized multiscale finite element method (GMsFEM). We present the construction of the offline and online multiscale basis functions. The offline multiscale basis functions are precomputed for the given heterogeneity and fracture network geometry, where for the construction, we solve a local spectral problem and use the dominant eigenvectors (appropriately defined) to construct multiscale basis functions. To construct the online basis functions, we use current information about the local residual and solve coupled poroelasticity problems in local domains. The online basis functions are used to enrich the offline multiscale space and rapidly reduce the error using residual information. Only with appropriate offline coarse-grid spaces can one guarantee a fast convergence of online methods. We present numerical results for poroelasticity problems in fractured and heterogeneous media. We investigate the influence of the number of offline and online basis functions on the relative errors between the multiscale solution and the reference (fine-scale) solution.https://www.mdpi.com/2311-5521/6/8/298multiscale methodGMsFEMporoelasticity problemfinite element methodheterogeneous mediafractured media |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Aleksei Tyrylgin Maria Vasilyeva Dmitry Ammosov Eric T. Chung Yalchin Efendiev |
spellingShingle |
Aleksei Tyrylgin Maria Vasilyeva Dmitry Ammosov Eric T. Chung Yalchin Efendiev Online Coupled Generalized Multiscale Finite Element Method for the Poroelasticity Problem in Fractured and Heterogeneous Media Fluids multiscale method GMsFEM poroelasticity problem finite element method heterogeneous media fractured media |
author_facet |
Aleksei Tyrylgin Maria Vasilyeva Dmitry Ammosov Eric T. Chung Yalchin Efendiev |
author_sort |
Aleksei Tyrylgin |
title |
Online Coupled Generalized Multiscale Finite Element Method for the Poroelasticity Problem in Fractured and Heterogeneous Media |
title_short |
Online Coupled Generalized Multiscale Finite Element Method for the Poroelasticity Problem in Fractured and Heterogeneous Media |
title_full |
Online Coupled Generalized Multiscale Finite Element Method for the Poroelasticity Problem in Fractured and Heterogeneous Media |
title_fullStr |
Online Coupled Generalized Multiscale Finite Element Method for the Poroelasticity Problem in Fractured and Heterogeneous Media |
title_full_unstemmed |
Online Coupled Generalized Multiscale Finite Element Method for the Poroelasticity Problem in Fractured and Heterogeneous Media |
title_sort |
online coupled generalized multiscale finite element method for the poroelasticity problem in fractured and heterogeneous media |
publisher |
MDPI AG |
series |
Fluids |
issn |
2311-5521 |
publishDate |
2021-08-01 |
description |
In this paper, we consider the poroelasticity problem in fractured and heterogeneous media. The mathematical model contains a coupled system of equations for fluid pressures and displacements in heterogeneous media. Due to scale disparity, many approaches have been developed for solving detailed fine-grid problems on a coarse grid. However, some approaches can lack good accuracy on a coarse grid and some corrections for coarse-grid solutions are needed. In this paper, we present a coarse-grid approximation based on the generalized multiscale finite element method (GMsFEM). We present the construction of the offline and online multiscale basis functions. The offline multiscale basis functions are precomputed for the given heterogeneity and fracture network geometry, where for the construction, we solve a local spectral problem and use the dominant eigenvectors (appropriately defined) to construct multiscale basis functions. To construct the online basis functions, we use current information about the local residual and solve coupled poroelasticity problems in local domains. The online basis functions are used to enrich the offline multiscale space and rapidly reduce the error using residual information. Only with appropriate offline coarse-grid spaces can one guarantee a fast convergence of online methods. We present numerical results for poroelasticity problems in fractured and heterogeneous media. We investigate the influence of the number of offline and online basis functions on the relative errors between the multiscale solution and the reference (fine-scale) solution. |
topic |
multiscale method GMsFEM poroelasticity problem finite element method heterogeneous media fractured media |
url |
https://www.mdpi.com/2311-5521/6/8/298 |
work_keys_str_mv |
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1721193402837499904 |