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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Bibliographic Details
Published in:Fluids (Basel) 2021-08, Vol.6 (8), p.298
Main Authors: Tyrylgin, Aleksei, Vasilyeva, Maria, Ammosov, Dmitry, Chung, Eric T., Efendiev, Yalchin
Format: Article
Language:English
Subjects:
Approximation
Basis functions
Boundary conditions
Composite materials
Construction
Eigenvalues
Eigenvectors
Error reduction
Finite element analysis
Finite element method
Finite volume method
fractured media
Geometry
GMsFEM
Heterogeneity
heterogeneous media
Mathematical models
Mechanics
multiscale method
Numerical methods
Poroelasticity
poroelasticity problem
Simulation
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Summary: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.
ISSN:2311-5521
2311-5521
DOI:10.3390/fluids6080298