Meshfree Generalized Multiscale Finite Element Method

Multiscale model reduction for shale gas transport in poroelastic fractured media

In this paper, we construct a class of Mixed Generalized Multiscale Finite Element Methods for the approximation on a coarse grid for an elliptic problem in thin two-dimensional domains. We consider the elliptic equation with homogeneous boundary conditions on the domain walls. For reference solution of the problem, we use a Mixed Finite Element Method on a fine grid that resolves complex geometry on the grid level. To construct a lower dimensional model, we use the Mixed Generalized Multiscale Finite Element Method, which is based on some multiscale basis functions for velocity fields. The construction of the basis functions is based on the local snapshot space that takes all possible flows on the interface between coarse cells into account. In order to reduce the size of the snapshot space and obtain the multiscale approximation, we solve a local spectral problem to identify dominant modes in the snapshot space. We present a convergence analysis of the presented multiscale method. Numerical results are presented for two-dimensional problems in three testing geometries along with the errors associated to different numbers of the multiscale basis functions used for the velocity field. Numerical investigations are conducted for problems with homogeneous and heterogeneous properties respectively.

In this work, we consider the elasticity problem in fractured media. For the efficient numerical solution, we present a Generalized Multiscale Finite Element Method (GMsFEM). GMsFEM is used for the construction of a coarse grid approximation of the problem by solution of the local spectral problems. We consider two types of the multiscale basis functions: (1) CG-GMsFEM with continuous multiscale basis functions and (2) DG-GMsFEM with discontinuous multiscale basis functions. The result of the numerical solution for the two-dimensional model problem is presented to show the performance of the presented multiscale method for fractured media. We compute error between the multiscale solution with the fine-scale solutions by choosing different numbers of multiscale basis functions.

In this paper we present and analyze a constraint energy minimizing generalized multiscale finite element method for convection diffusion equations. To define the multiscale basis functions, we first build an auxiliary multiscale space by solving local spectral problems motivated by analysis. Then a constraint energy minimization performed in the oversampling domains is exploited to construct the multiscale space. The resulting multiscale basis functions have a good decay property even for high contrast diffusion and convection coefficients. Furthermore, if the number of oversampling layers is chosen properly, we can prove that the convergence rate is proportional to the coarse meshsize. Our analysis also indicates that the size of the oversampling domain weakly depends on the contrast of the heterogeneous coefficients. Several numerical experiments are presented illustrating the performance of our method. © 2023 Society for Industrial and Applied Mathematics Publications. All rights reserved.

The paper explores the dual-continuum unsaturated filtration model in heterogeneous porous media with rough surface topography. The rough surface topography plays a crucial role in determining the flow process and incorporates multiscale features. The mathematical model is based on the Richard’s equation with a Robin boundary condition for the rough surface boundary. The Online Generalized Multiscale Finite Element Method (Online GMsFEM) is used for coarse-grid discretization. Multiscale basis functions that encompass small-scale heterogeneities are created. Additional basis functions are generated to accommodate the impact of boundary conditions on rough surfaces. We present a consrtuction of online mulsticale basis functions that based on local residuals. All multiscale basis functions are coupled and take into account transfer term between continuums. Numerical results for a two-dimensional problem are provided, and relative errors between multiscale and reference solutions are calculated for varying numbers of multiscale basis functions to verify the results. The influence of online bases on the algorithm’s accuracy is also discussed.

In the reservoir simulation, mathematical modeling of the fluid flow and geo - mechanics in the fractured porous media plays an important role. Fracture networks have complex geometries, exist in the multiple scales and typically have very small thickness compared to typical reservoir sizes. Due to high permeability, fractures have a significant impact on the flow processes. In this work, we consider a discrete fracture model for coupled flow and mechanics problems. We construct coarse grid approximation using Generalized Multiscale Finite Element method (GMsFEM). In this method, we solve local spectral problems for construction to the multiscale basis functions for pressure and displacements. We present numerical results for two - dimensional model problem.

In this paper, we consider a coupled system of equations that describes simplified magnetohydrodynamics (MHD) problem in perforated domains. We construct a fine grid that resolves the perforations on the grid level in order to use a traditional approximation. For the solution on the fine grid, we construct approximation using the mixed finite element method. To reduce the size of the fine grid system, we will develop a Mixed Generalized Multiscale Finite Element Method (Mixed GMsFEM). The method differs from existing approaches and requires some modifications to represent the flow and magnetic fields. Numerical results are presented for a two-dimensional model problem in perforated domains. This model problem is a special case for the general 3D problem. We study the influence of the number of multiscale basis functions on the accuracy of the method and show that the proposed method provides a good accuracy with few basis functions.

This paper presents the Meshfree Generalized Multiscale Finite Element Method (MFGMsFEM) for solving parabolic problems in multicontinuum media. Multicontinuum models are widely used in various fields, such as petroleum engineering, hydrogeology, and biomechanics, to describe fluid flow in fractured porous materials. The proposed approach leverages the advantages of both mesh free methods and the Generalized Multiscale Finite Element Method (GMsFEM), allowing for efficient and accurate numerical modeling. The novelty of this work lies in the development of multiscale basis functions, constructed using both split and coupled strategies. Numerical experiments demonstrate the effectiveness of the method in capturing complex flow dynamics while significantly reducing computational costs compared to traditional fine-grid simulations. The results show that the coupled multiscale basis functions provide higher accuracy in modeling flow interactions within multicontinuum media.

An adaptive local–global multiscale finite volume element method for two-phase flow simulations

Constraint energy minimizing generalized multiscale discontinuous Galerkin method

Conventional finite-element method for solving Helmholtz equation on highly heterogeneous media usually require very finely discretized mesh to accurately represent the medium property variations, and therefore is computationally expensive. We present a continuous Galerkin generalized multiscale finite element method for solving Helmholtz equation in media with variable velocity and mass density on coarse mesh. Instead of using polynomial basis functions, our multiscale method use multiscale functions obtained by multiplying the eigenfunctions from local spectral problems with an appropriately selected multiscale partition of unity to form the approximation space on the coarse mesh. These multiscale basis can efficiently capture the information of fine scale features of the medium without solving the problem on the fine mesh. We show by a numerical example that our multiscale method can greatly reduce the dimension of the coefficient matrix of Helmholtz equation system, as well as the computational time, with very small error.

Constraint Energy Minimizing Generalized Multiscale Finite Element Method

A hybridizable discontinuous Galerkin Generalized Multiscale Finite element method for highly heterogeneous linear elasticity problems

Seismic data are routinely used to infer in situ properties of earth materials on many scales, ranging from global studies to investigations of surficial geological formations. While inversion and imaging algorithms utilizing these data have improved steadily, there are remaining challenges that make detailed measurements of the properties of some geologic materials very difficult. For example, the determination of the concentration and orientation of fracture systems is prohibitively expensive to simulate on the fine grid and, thus, some type of coarse-grid simulations are needed. In this paper, we describe a new multiscale finite element algorithm for simulating seismic wave propagation in heterogeneous media. This method solves the wave equation on a coarse grid using multiscale basis functions and a global coupling mechanism to relate information between fine and coarse grids. Using a mixed formulation of the wave equation and staggered discontinuous basis functions, the proposed multiscale methods have the following properties. • The total wave energy is conserved. • Mass matrix is diagonal on a coarse grid and explicit energy-preserving time discretization does not require solving a linear system at each time step. • Multiscale basis functions can accurately capture the subgrid variations of the solution and the time stepping is performed on a coarse grid. We discuss various subgrid capturing mechanisms and present some preliminary numerical results.

A two-stage ensemble Kalman filter based on multiscale model reduction for inverse problems in time fractional diffusion-wave equations