Analyse av kontrollvolummetoder - Analyse av kontrollvolummetoder - Biannual report 2009

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Biannual report VISTA 2009

Project title Analyse av kontrollvolummetoder

Project director: Aavatsmark, Ivar, Centre for Integrated Petroleum Research (CIPR)

Post-doc/ scholar: Stephansen, Annette

Project duration: 01.09.08-30.04.11

Technical contact person in Statoil: Høier, Lars

Division head: Amundsen, Lasse

Project number: 6344

Object:

The object of this project is to analyse control volume methods and in particular the methods known as multi-point flux approximation (MPFA) methods. The MPFA methods have been extensively tested numerically, but the mathematical analysis is less developed. In particular, one would like to establish the convergence properties of the various MPFA methods on rough grids, i.e. non-orthogonal grids, and on general (polyhedral) grids. The convergence of the MPFA method known as the O-method has been established in 2d on rough grids, but the analysis is not easily extended into 3d. The convergence of the L-method on rough grids has not been proved even in 2d. The influence of the heterogeneities and anisotropies of the permeability tensor is also an important aspect to investigate. The specific aim of the mathematical analysis is to establish criteria that can be used to create grids which guarantee that the method converges. A theoretical analysis will also highlight the mathematical differences between the different MPFA methods currently in use and establish their relative merits.

Status:

Pursuing the similarities between the mimetic finite difference (MFD) method and the MPFA methods, we have established a convergence proof of the MPFA O-method for general grids in 2D and 3D. The proof relies on certain assumptions which concern the grid and the permeability tensor, and we have thus examined what deformations and what anisotropies are permissible in order for the convergence proof still to be valid. We have also focused on the optimality of the proof, noting that convergence in a different norm might be possible even when the assumption for convergence is violated. Our next focus has been on the L-method where the first step has been to establish the corresponding quadrature when going from a dual-grid L-method definition to a definition on the original grid. The quadrature has permitted us to adapt the convergence proof for the MPFA O-method on general grids in 3D with some slight modifications. The results have been presented at the SIAM conference on GS09 in Leipzig. Articles are under preparation.