27 research outputs found
Computational Multiscale Methods for Linear Poroelasticity with High Contrast
In this work, we employ the Constraint Energy Minimizing Generalized
Multiscale Finite Element Method (CEM-GMsFEM) to solve the problem of linear
heterogeneous poroelasticity with coefficients of high contrast. The proposed
method makes use of the idea of energy minimization with suitable constraints
in order to generate efficient basis functions for the displacement and the
pressure. These basis functions are constructed by solving a class of local
auxiliary optimization problems based on eigenfunctions containing local
information on the heterogeneity. Techniques of oversampling are adapted to
enhance the computational performance. Convergence of first order is shown and
illustrated by a number of numerical tests.Comment: 14 pages, 9 figure
Non-Local Multi-Continuum method (NLMC) for Darcy-Forchheimer flow in fractured media
This work presents the application of the non-local multicontinuum method
(NLMC) for the Darcy-Forchheimer model in fractured media. The mathematical
model describes a nonlinear flow in fractured porous media with a high inertial
effect and flow speed. The space approximation is constructed on the
sufficiently fine grid using a finite volume method (FVM) with an embedded
fracture model (EFM) to approximate lower dimensional fractures. A non-local
model reduction approach is presented based on localization and constraint
energy minimization. The multiscale basis functions are constructed in
oversampled local domains to consider the flow effects from neighboring local
domains. Numerical results are presented for a two-dimensional formulation with
two test cases of heterogeneity. The influence of model nonlinearity on the
multiscale method accuracy is investigated. The numerical results show that the
non-local multicontinuum method provides highly accurate results for
Darcy-Forchheimer flow in fractured media
A Generalized Multiscale Finite Element Method for poroelasticity problems II: nonlinear coupling
In this paper, we consider the numerical solution of some nonlinear poroelasticity problems that are of Biot type and develop a general algorithm for solving nonlinear coupled systems. We discuss the difficulties associated with flow and mechanics in heterogenous media with nonlinear coupling. The central issue being how to handle the nonlinearities and the multiscale scale nature of the media. To compute an efficient numerical solution we develop and implement a Generalized Multiscale Finite Element Method (GMsFEM) that solves nonlinear problems on a coarse grid by constructing local multiscale basis functions and treating part of the nonlinearity locally as a parametric value. After linearization with a Picard Iteration, the procedure begins with construction of multiscale bases for both displacement and pressure in each coarse block by treating the staggered nonlinearity as a parametric value. Using a snapshot space and local spectral problems, we construct an offline basis of reduced dimension. From here an online, parametric dependent, space is constructed. Finally, after multiplying by a multiscale partitions of unity, the multiscale basis is constructed and the coarse grid problem then can be solved for arbitrary forcing and boundary conditions. We implement this algorithm on a geometry with a linear and nonlinear pressure dependent permeability field and compute error between the multiscale solution with the fine-scale solutions