An Online Generalized Multiscale Finite Element Method for Unsaturated Filtration Problem in Fractured Media

Denis Spiridonov, Maria Vasilyeva, Aleksei Tyrylgin and Eric T. Chung
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Denis Spiridonov: Multiscale Model Reduction Laboratory, North-Eastern Federal University, 677980 Yakutsk, Russia
Maria Vasilyeva: Center of Innovation for Flow through Porous Media, University of Wyoming, Laramie, WY 82071, USA
Aleksei Tyrylgin: Multiscale Model Reduction Laboratory, North-Eastern Federal University, 677980 Yakutsk, Russia
Eric T. Chung: Department of Mathematics, The Chinese University of Hong Kong, Shatin, New Territories, Hong Kong, China

Mathematics, 2021, vol. 9, issue 12, 1-14

Abstract: In this paper, we present a multiscale model reduction technique for unsaturated filtration problem in fractured porous media using an Online Generalized Multiscale finite element method. The flow problem in unsaturated soils is described by the Richards equation. To approximate fractures we use the Discrete Fracture Model (DFM). Complex geometric features of the computational domain requires the construction of a fine grid that explicitly resolves the heterogeneities such as fractures. This approach leads to systems with a large number of unknowns, which require large computational costs. In order to develop a more efficient numerical scheme, we propose a model reduction procedure based on the Generalized Multiscale Finite element method (GMsFEM). The GMsFEM allows solving such problems on a very coarse grid using basis functions that can capture heterogeneities. In the GMsFEM, there are offline and online stages. In the offline stage, we construct snapshot spaces and solve local spectral problems to obtain multiscale basis functions. These spectral problems are defined in the snapshot space in each local domain. To improve the accuracy of the method, we add online basis functions in the online stage. The construction of the online basis functions is based on the local residuals. The use of online bases will allow us to get a significant improvement in the accuracy of the method. We present results with different number of offline and online multisacle basis functions. We compare all results with reference solution. Our results show that the proposed method is able to achieve high accuracy with a small computational cost.

Keywords: fractured media; unsaturated filtration; Richards equation; multiscale finite element method; online generalized multiscale finite element method; multiscale model reduction (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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