Analysis of Control Volume Heterogeneous Multiscale Methods for Single Phase Flow in Porous Media

Alyaev, Sergey; Keilegavlen, Eirik; Nordbotten, Jan Martin

doi:10.1137/120885541

Cited by 15 publications

(20 citation statements)

References 26 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Using here the variant of Stokes problem (12), which excludes pressure, we can restate (27) as follows. For any x ∈ {Q H } and i ∈ {1, .…”

Section: Uniform Ellipticity Of a H (X)mentioning

confidence: 99%

See 1 more Smart Citation

An Adaptive Finite Element Heterogeneous Multiscale Method for Stokes Flow in Porous Media

Abdulle¹,

Budáč²

2015

Multiscale Model. Simul.

View full text Add to dashboard Cite

A finite element heterogeneous multiscale method is proposed for solving the Stokes problem in porous media. The method is based on the coupling of an effective Darcy equation on a macroscopic mesh, with unknown permeabilities recovered from micro finite element calculations for Stokes problems on sampling domains centered at quadrature points in each macro element. The numerical method accounts for non-periodic microscopic geometry that can be obtained from a smooth deformation of a reference pore sampling domain. The computational work is nevertheless independent of the smallness of the pore structure. A priori error estimates reveal that the overall accuracy of the numerical scheme is limited by the regularity of the solutions of the Stokes micro problems. This regularity is low for a typical situation of non-convex microscopic pore geometries. We therefore propose an adaptive scheme with micro-macro mesh refinement driven by residual-based indicators that quantify both the macro and micro errors. A posteriori error analysis is derived for the new method. Two and three dimensional numerical experiments confirm the robustness and the accuracy of the adaptive method.Keywords. Stokes flow, Darcy equation, numerical homogenization, a posteriori error estimates, adaptive finite element method AMS subject classifications. 65N30, 76D07, 74Qxx, 35Q861. Introduction. Fluid flow through porous media is an important process appearing in a wide range of engineering and technical applications. It is present in the modeling of subsurface contamination and filtration, textile properties, biomedical materials, or natural reservoirs [30,50,55,56]. The length-scale of a porous structure is usually much smaller than the computational domain of interest. Standard numerical methods relying on discretization of the porous domain, such as the finite element method (FEM), need to resolve the finest scale of the geometry, which is denoted by ε in what follows. Such techniques often lead to numerical problems of prohibitive size and computational cost.In practical applications one is often interested in macroscopic quantities such as bulk properties of the fluid flow. Mathematical models describing such macroscopic quantities are based on averaging techniques such as homogenization. The derivation of effective equations of flow in porous media can be traced back to Darcy [28]. Rigorous homogenization theory of Stokes flow in periodic porous media appeared first in [44] with a proof of convergence by Tartar [47]. This proof was generalized by Allaire [10] to allow for connected solid porous structures in three dimensions. The effective pressure is given by an elliptic (Darcy) equation which contains the effective permeability tensor that depends on the pore geometry and can be computed using the so-called Stokes micro problems. The homogenization theory was further expanded by introducing correctors and ε-dependent error estimates [34] (see also [21]) and to random stochastically homogeneous media [15].Numerical multiscale algo...

show abstract

“…Using here the variant of Stokes problem (12), which excludes pressure, we can restate (27) as follows. For any x ∈ {Q H } and i ∈ {1, .…”

Section: Uniform Ellipticity Of a H (X)mentioning

confidence: 99%

“…A priori analysis or convergence rates of this method were not discussed. The multiscale approach presented in [12] uses control volume method to discretize the Darcy macro equation. The reduced regularity of non-convex micro domains is taken into account and an appropriate estimate of the micro error is derived.…”

mentioning

confidence: 99%

An Adaptive Finite Element Heterogeneous Multiscale Method for Stokes Flow in Porous Media

Abdulle¹,

Budáč²

2015

Multiscale Model. Simul.

View full text Add to dashboard Cite

show abstract

“…Other numerical methods that rely on the Stokes equation at pore scale have been proposed in [10,13,23]. The multiscale approach in [10] uses the control volume method to discretize the Darcy equation and a Navier-Stokes model is used on the micro scale.…”

Section: Introductionmentioning

confidence: 99%

“…The multiscale approach in [10] uses the control volume method to discretize the Darcy equation and a Navier-Stokes model is used on the micro scale. While this approach allows a conservative macro solver and more precise micro model, there are some limiting assumptions: piecewise periodic micro structure, alignment of the micro structure to a coarse grid, and no volumetric forces.…”

Section: Introductionmentioning

confidence: 99%

A Discontinuous Galerkin Reduced Basis Numerical Homogenization Method for Fluid Flow in Porous Media

Abdulle¹,

Budáč²

2017

SIAM J. Sci. Comput.

View full text Add to dashboard Cite

We present a new conservative multiscale method for Stokes flow in heterogeneous porous media. The method couples a discontinuous Galerkin finite element method (DG-FEM) at the macroscopic scale for the solution of an effective Darcy equation with a Stokes solver at the pore scale to recover effective permeabilities at macroscopic quadrature points. To avoid costly computation of numerous Stokes problems throughout the macroscopic computational domain, the pore geometry is parametrized and a model order reduction algorithm is used to select representative microscopic geometries. Accurate Stokes solutions and related permeabilities are obtained for these representative geometries in an offline stage. In an online stage, the DG-FEM is computed with permeabilities recovered at the desired macroscopic quadrature points from the precomputed Stokes solutions. The multiscale method is shown to be mass conservative at the macro scale and the computational cost for the online stage is similar to the cost of solving a single scale Darcy problem. Numerical experiments for two and three dimensional problems illustrate the efficiency and the performance of the proposed method.

show abstract

“…Other adaptive algorithms for the HMM focusing only on the macroscopic elliptic problem can be found in [1,2,7,32]. Other numerical method that rely on the Stokes equation at pore scale have been proposed in [2,10,18,41]. The multiscale approach in [10] uses the control volume method to discretize the Darcy equation, while the multiscale FEM [18] uses a hierarchy of macroscopic grids where micro problems are solved with various accuracy.…”

Section: Introductionmentioning

confidence: 99%

A reduced basis finite element heterogeneous multiscale method for Stokes flow in porous media

Abdulle

Budáč

2016

Computer Methods in Applied Mechanics and Engineering

View full text Add to dashboard Cite

A reduced basis Darcy-Stokes finite element heterogeneous multiscale method (RB-DS-FE-HMM) is proposed for the Stokes problem in porous media. The multiscale method is based on the Darcy-Stokes finite element heterogeneous multiscale method (DS-FE-HMM) introduced in [A. Abdulle, O. Budáč, Multiscale Model. Simul. 13 (2015)] that couples a Darcy equation solved on a macroscopic mesh, with missing permeability data extracted from the solutions of Stokes micro problems at each macroscopic quadrature point. To overcome the increasingly growing cost of repeatedly solving the Stokes micro problems as the macroscopic mesh is refined, we parametrize the microscopic solid geometry and approximate the infinite-dimensional manifold of parameter dependent solutions of Stokes problems by a low-dimensional space. This low-dimensional (reduced basis) space is obtained in an offline stage by a greedy algorithm and used in an online stage to compute the effective Darcy permeability at a cost independent of the microscopic mesh. The discretization of the parametrized Stokes problems relies on a Petrov-Galerkin formulation that allows for a stable and fast online evaluation of the required permeabilities. A priori and a posteriori estimates of the RB-DS-FE-HMM are derived and a residual-based adaptive algorithm is proposed. Two-and three-dimensional numerical experiments confirm the accuracy of the RB-DS-FE-HMM and illustrate the speedup compared to the DS-FE-HMM.

show abstract

Analysis of Control Volume Heterogeneous Multiscale Methods for Single Phase Flow in Porous Media

Cited by 15 publications

References 26 publications

An Adaptive Finite Element Heterogeneous Multiscale Method for Stokes Flow in Porous Media

An Adaptive Finite Element Heterogeneous Multiscale Method for Stokes Flow in Porous Media

A Discontinuous Galerkin Reduced Basis Numerical Homogenization Method for Fluid Flow in Porous Media

A reduced basis finite element heterogeneous multiscale method for Stokes flow in porous media

Contact Info

Product

Resources

About