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

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.

Prediction of numerical homogenization using deep learning for the Richards equation

“…More specifically, at this online stage, let V j = H 1 0 (w j ) ∩ V h . Using the procedure in [55,28], which is derived from the residual-based online basis enrichment for GMsFEM [13], our online multiscale basis function η n j,s+1 ∈ V j is the solution to the following local problem in w j :…”

“…Moreover, for nonlinear problems, the online GMsFEM is an appropriate choice, as it introduces additional online multiscale basis functions to include any time-dependent changes in the problem's properties [13,15,55]. The online GMsFEM showed good performance in one of the authors' previous work [55], where one can observe superiority of the online GMsFEM over the traditional offline GMsFEM, in solving the Richards equation describing unsaturated infiltration. Within our paper, for nonlinear single-continuum Richards equation, we combine the online GMsFEM with the idea of Picard iterative process [28].…”

“…Denis Spiridonov ; a Laboratory of Computational Technologies for Modeling Multiphysical and Multiscale Permafrost Processes, North-Eastern Federal University, 677000 Yakutsk, Republic of Sakha (Yakutia), Russia; d.stalnov@mail.ru Sergei Stepanov ; a Laboratory of Computational Technologies for Modeling Multiphysical and Multiscale Permafrost Processes, North-Eastern Federal University, 677000 Yakutsk, Republic of Sakha (Yakutia), Russia; cepe2a@inbox.ru the work with offline multiscale basis functions [54], and the work utilizing online multiscale basis functions [55]. More specifically, in the offline GMsFEM (as in [54]), multiscale basis functions (offline function space) are generated and then employed to solve the Richards equation on a coarse grid.…”

An Online Generalized Multiscale finite element method for heat and mass transfer problem with artificial ground freezing

“…A multiscale method that satisfies the properties of mass conservation is called mixed multiscale finite element method (Mixed MsFEM) [35][36][37]. An online generalized multiscale finite element method (Online GMsFEM) [38][39][40][41] is particularly suited for nonlinear problems because it executes the procedure of enriching a multiscale space during the online stage of the method. A special type of multiscale basis functions based on constrained energy minimization problems are developed in [42][43][44][45] and well-known as nonlocal multicontinuum method(NLMC).…”

Mixed Generalized Multiscale Finite Element Method for Darcy-Forchheimer Model

“…In the constraint energy minimizing generalized multiscale finite element method (CEM-GMsFEM), the multiscale basis functions are constructed in oversampled local domains and therefore take into account the influence of heterogeneity in neighboring local domains [52][53][54][55][56][57]. For nonlinear problems, the online generalized multiscale finite method (Online GMsFEM) can be used where additional multiscale bases take into account changes in properties in nonlinear problems [58][59][60][61]. In our previous work [62], we used the mixed generalized finite element method for the Darcy-Fochheimer model in a heterogeneous domain.…”