Analyse av kontrollvolummetoder - Annual report VISTA 2009

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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. Our first aim has been to analyse the convergence of the MPFA methods on rough grids, i.e. non-orthogonal grids, and on general (polyhedral) grids. The specific aim of the mathematical analysis is to establish criteria that can be used to create grids which guarantee that the method converges. The theoretical analysis also gives a practical tool to choose between the L-method and the O-method given a specific grid and a specific distribution of anisotropy. Future work will look at interpolations at polyhedra that reproduce uniform flow (the same property that is one of the building blocks of the MPFA methods) and MPFA multiscale methods.

Status:

The strategy adapted so far has been to look at the similarities between the MPFA methods and the mimetic finite difference (MFD) method, since there exists a convergence proof for the latter on general polyhedral grids in 3d. Both methods have the characteristic that they give exact result for linear pressure fields and refrain from solving the velocity field in the interior of the elements. The MPFA methods calculates the flux explicitly based on local pressure differences, and this gives rise to the non-symmetry of the method. The non-symmetry in turn leads to problems when dealing with very large anisotropies or grid deformations. In the article on the O-method (submitted) we specifically look at the limits on anisotropy and grid deformation that are assumed in the convergence proof which we propose. In the article on the L-method the analysis show that by choosing the L-stencils opportunely, the method will converge in cases with high anisotropy where the convergence proof for the O-method fails. However, the restrictions on mesh conformity are more severe. We note that the proof for the L-method is only valid for fixed stencils, while the original L-method allows the user to choose the L-stencil for any flux independently of the stencils used for neighbouring fluxes. Some of these results were presented at the SIAM Conference on Mathematical & Computational Issues in the Geosciences in Leipzig, June 15-18 (SIAM GS 09).

Publications:

Klausen, Runhild and Stephansen, Annette. Convergence of the MPFA O-method on general grids. Submitted.

Stephansen, Annette. Convergence of the MPFA L-method on general grids. Submitted.