Skip to main content
OSTI.GOVU.S. Department of Energy Office of Scientific and Technical Information
U.S. Department of Energy
Office of Scientific and Technical Information
 

Numerical Multilevel Upscaling for Incompressible Flow in Reservoir Simulation: An Element-Based Algebraic Multigrid (AMGe) Approach

Journal Article · · SIAM Journal on Scientific Computing
DOI:https://doi.org/10.1137/140988991· OSTI ID:1461736
 [1];  [2];  [3];  [4]
  1. Technical Univ. of Denmark, Lyngby (Denmark); Technical Univ. of Denmark, Lloyd's Register Consulting, Hellerup (Denmark)
  2. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States); The Univ. of Texas at Austin, Austin, TX (United States)
  3. Technical Univ. of Denmark, Lyngby (Denmark)
  4. Lawrence Livermore National Lab. (LLNL), Livermore, CA (United States)
+ Show Author Affiliations
Here, we study the application of a finite element numerical upscaling technique to the incompressible two-phase porous media total velocity formulation. Specifically, an element-agglomeration-based algebraic multigrid (AMGe) technique with improved approximation properties is used, for the first time, to generate upscaled and accurate coarse systems for the reservoir simulation equations. The upscaling technique is applied to both the mixed system for velocity and pressure and to the hyperbolic transport equations, providing fully upscaled systems. By introducing additional degrees of freedom associated with nonplanar interfaces between agglomerates, the coarse velocity space has guaranteed approximation properties. The employed AMGe technique provides coarse spaces with desirable local mass conservation and stability properties analogous to the original pair of Raviart--Thomas and piecewise discontinuous polynomial spaces, resulting in strong mass conservation for the upscaled systems. Due to the guaranteed approximation properties and the generic nature of the AMGe method, recursive multilevel upscaling is automatically obtained. Furthermore, this technique works for both structured and unstructured meshes. Multiscale mixed finite elements exhibit accuracy for general unstructured meshes but do not in general lead to nested hierarchy of spaces. Multiscale multilevel mimetic finite differences generate nested spaces but lack the adaptivity of the flux representation on coarser levels that the proposed AMGe approach offers. Thus, the proposed approach can be seen as a rigorous bridge that merges the best properties of these two existing methods. The accuracy and stability of the studied multilevel AMGe upscaling technique is demonstrated on two challenging test cases.
Research Organization:
Lawrence Livermore National Laboratory (LLNL), Livermore, CA (United States)
Sponsoring Organization:
USDOE National Nuclear Security Administration (NNSA)
Grant/Contract Number:
AC52-07NA27344
OSTI ID:
1461736
Report Number(s):
LLNL-JRNL--661295; 782221
Journal Information:
SIAM Journal on Scientific Computing, Journal Name: SIAM Journal on Scientific Computing Journal Issue: 1 Vol. 39; ISSN 1064-8275
Publisher:
SIAMCopyright Statement
Country of Publication:
United States
Language:
English

References (41)

Element agglomeration coarse Raviart-Thomas spaces with improved approximation properties: COARSE RAVIART-THOMAS SPACES WITH IMPROVED APPROXIMATION PROPERTIES journal January 2012
Sparse matrix element topology with application to AMG(e) and preconditioning journal January 2002
On two-grid convergence estimates
  • Falgout, Robert D.; Vassilevski, Panayot S.; Zikatanov, Ludmil T.
  • Numerical Linear Algebra with Applications, Vol. 12, Issue 5-6 https://doi.org/10.1002/nla.437
journal January 2005
On some versions of the element agglomeration AMGe method journal September 2008
Preconditioning discretizations of systems of partial differential equations journal April 2010
A Multiscale Finite Element Method for Elliptic Problems in Composite Materials and Porous Media journal June 1997
Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients journal January 2011
An element agglomeration nonlinear additive Schwarz preconditioned Newton method for unstructured finite element problems journal June 2005
Multiscale mixed/mimetic methods on corner-point grids journal January 2008
A discontinuous Galerkin method for two-phase flow in a porous medium enforcing H(div) velocityand continuous capillary pressure journal September 2013
Multi-scale finite-volume method for elliptic problems in subsurface flow simulation journal May 2003
Mixed multiscale finite elements and streamline methods for reservoir simulation of large geomodels journal March 2005
An adaptive local–global multiscale finite volume element method for two-phase flow simulations journal March 2007
A multilevel multiscale mimetic (M3) method for two-phase flows in porous media journal July 2008
Implementation of higher-order methods for robust and efficient compositional simulation journal April 2010
A multiscale two-point flux-approximation method journal October 2014
A multiscale multilevel mimetic (M3) method for well-driven flows in porous media journal May 2010
Numerical solution of saddle point problems journal April 2005
An Improved IMPES Method for Two-Phase Flow in Porous Media journal March 2004
Multilevel upscaling through variational coarsening: MULTILEVEL VARIATIONAL UPSCALING journal February 2006
Compositional modeling of three-phase flow with gravity using higher-order finite element methods: COMPOSITIONAL MODELING OF 3-PHASE FLOW journal May 2011
A mixed multiscale finite element method for elliptic problems with oscillating coefficients journal June 2002
On the Use of a Mixed Multiscale Finite Element Method for GreaterFlexibility and Increased Speed or Improved Accuracy in Reservoir Simulation journal January 2004
Subgrid Upscaling and Mixed Multiscale Finite Elements journal January 2006
A Hierarchical Multiscale Method for Two-Phase Flow Based upon Mixed Finite Elements and Nonuniform Coarse Grids journal January 2006
Multigrid Methods for PDE Optimization journal May 2009
Exact de Rham Sequences of Spaces Defined on Macro-Elements in Two and Three Spatial Dimensions journal January 2008
Adaptive Strategies in the Multilevel Multiscale Mimetic (M3) Method for Two-Phase Flows in Porous Media journal July 2011
A Block-Diagonal Algebraic Multigrid Preconditioner for the Brinkman Problem journal January 2013
A Mixed Formulation for the Brinkman Problem journal January 2014
Two-Level Adaptive Algebraic Multigrid for a Sequence of Problems with Slowly Varying Random Coefficients journal January 2013
A Hierarchical Multilevel Markov Chain Monte Carlo Algorithm with Applications to Uncertainty Quantification in Subsurface Flow journal January 2015
Analysis of a Two-Scale, Locally Conservative Subgrid Upscaling for Elliptic Problems journal January 2004
Spectral AMGe ($\rho$AMGe) journal January 2003
A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs journal January 1998
Matrix-Dependent Multigrid Homogenization for Diffusion Problems journal January 1998
AMGE Based on Element Agglomeration journal January 2001
A Generalized Convection-Diffusion Model for Subgrid Transport in Porous Media journal January 2003
Coarse Spaces by Algebraic Multigrid: Multigrid Convergence and Upscaling Error Estimates journal April 2011
The Construction of the Coarse de Rham Complexes with Improved Approximation Properties journal January 2014
Interpretation of Well-Block Pressures in Numerical Reservoir Simulation With Nonsquare Grid Blocks and Anisotropic Permeability journal June 1983

Cited By (2)

Scalable hierarchical PDE sampler for generating spatially correlated random fields using nonmatching meshes: Scalable hierarchical PDE sampler using nonmatching meshes journal January 2018
Scalable hierarchical PDE sampler for generating spatially correlated random fields using non-matching meshes preprint January 2017

Similar Records

Upscaling of Mixed Finite Element Discretization Problems by the Spectral AMGe Method
Journal Article · Wed Sep 21 20:00:00 EDT 2016 · SIAM Journal on Scientific Computing · OSTI ID:1368024
NONLINEAR MULTIGRID SOLVER EXPLOITING AMGe COARSE SPACES WITH APPROXIMATION PROPERTIES
Technical Report · Thu Jan 21 23:00:00 EST 2016 · OSTI ID:1237574

Related Subjects

97 MATHEMATICS AND COMPUTING
discontinuous Galerkin finite element method
element-based algebraic multigrid
mixed finite element method
multilevel upscaling
numerical upscaling
porous media flow
reservoir simulation
subsurface flow