Annual report 2008

Annual report 2008 of the VISTA professorship

Research program

The activities of this project are part of a larger program where the objective is to establish
recommendations for reservoir engineers and industrial simulator developers regarding
the choice of grid and associated discretization used in reservoir simulation, especially in
regions of complex geological structures and near-well regions. Experience with different
grids in regions with high flow density shows that this is an important and unsolved issue,
especially for multiphase flow.

The objectives of the project are to investigate important properties, such as convergence rate
and monotonicity, for control-volume methods on general grids in reservoir simulation.
Although test runs will be done for multiphase flow, the properties will mainly be investigated
for single-phase flow. Satisfactory convergence properties for single-phase flow are a
prerequisite for proper handling of multiphase flow, while monotonicity of the method is most
important for multiphase flow. Special emphasis lies on the robustness of the methods with
respect to anisotropy and cell geometry. Both two-point and multipoint flux approximation
(MPFA) methods are investigated.

Activities in 2008

Solutions of control-volume methods may experience spurious oscillations, even if the
solution converges to the correct solution. Also, at no-flow boundaries, spurious extrema may
occur. Such spurious oscillations and such no-flow boundary extrema only occur for
nonmonotone methods. Unfortunately, multipoint flux approximation (MPFA) methods are
only conditionally monotone, and monotonicity conditions seems to be more restrictive than
convergence conditions.

For two-dimensional triangular grids, an MPFA method which was optimal with respect to
convergence robustness, was presented at MAFELAP, London, in 2006. Many MPFA
methods may be constructed for triangular grids, and the most natural of these have been
compared with respect to monotonicity. The result of this investigation is that the method
which is optimal with respect to convergence is also optimal with respect to monotonicity.
This is an unexpected, but very nice, result.

Monotonicty tests have also been performed for 3D quadrilateral grids. Here, the
monotonicity investigations show that the monotonicity requirement may be quite restrictive.
The analysis indicates that better methods may be constructed. Further work is therefore
needed for 3D cases.

The MPFA methods have been tested in near-well regions, and there the results are quite
positive. Traditional two-point flux methods do not converge for near-well flow in anisotropic
media, and MPFA seems to be the only realistic alternative. It was expected that a radial grid
would be optimal, but this is not the case. A triangular grid based on a logarithmic spacing of
the grid points seems to be optimal. Due to the logarithmic spacing, monotonicity is hardly an
issue, at least not in 2D.

The near-well grids used in 2D may be extended to 3D via a prismatic extension. Based on
the two- and three-dimensional tests described above, it is not clear how well the prismatic 3D
grid will behave with respect to monotonicity. There may be restictions on the hight of the
prisms, but these restrictions are presently not known.

Other activities

In September 2008, the 11th European Conference on the Mathematics of Oil Recovery
(ECMOR XI) was arranged in Bergen with Ivar Aavatsmark as chairman. The conference
attracted 170 participants from 16 countries. There were 88 presentations. Perhaps
coincidentally, the topic with the largest number of contributions was multipoint flux
approximation (MPFA) methods.

Supervision/teaching

PhD student Sissel Mundal: MPFA discretization in near well region.
PhD student Eirik Keilegavlen: Monotonicity of MPFA methods on unstructured grids.
PhD student Andreas Sandvin: Properties of multiscale methods.
Post doc Annette Stephansen: Analysis of control volume methods.
Post doc Radinka Yorgova: Gridding.
Post doc Elsa Moggia: Compositional simulation in near-critical regions.

Completed papers 2008

1. U. Ölmann, I. Aavatsmark, B. Flemisch, R. Helmig: Tensorial relative permeabilities
and their treatment with multipoint flux approximation, Computational Geosciences,
Submitted 2008.
2. E. Keilegavlen, J. M. Nordbotten, I. Aavatsmark: Sufficient criteria are also necessary
for monotone control volume methods, Applied Mathematics Letters, Submitted 2008.
3. I. Aavatsmark, G. T. Eigestad, B.-O. Heimsund, B. T. Mallison, J. M. Nordbotten, E.
Øian: A new finite volume approach to efficient discretization on challenging grids,
SPE Journal, Submitted 2008.
Published papers 2008
1. S. K. Khattri, I. Aavatsmark: Numerical convergence on adaptive grids for control
volume methods, Numerical Methods for Partial Differential Equations, 24:465–475,
2008.
2. D. Eydinov, S. I. Aanonsen, J. Haukås, I. Aavatsmark: A method for automatic history
matching of a compositional reservoir simulator with multipoint flux approximation,
Computational Geosciences, 12:209–225, 2008.
3. I. Aavatsmark: Comparison of monotonicity for some multipoint flux approximation
methods, In: R. Eymard and J.-M. Hérard, editors, Finite Volumes for Complex
Applications V, pages 19–34, Wiley-ISTE, London, 2008.
4. S. Mundal, D. A. Di Pietro, I. Aavatsmark: Benchmark on anisotropic problems:
Compact-stencil MPFA method for heterogeneous highly anisotropic second-order
problems, In: R. Eymard and J.-M. Hérard, editors, Finite Volumes for Complex
Applications V, pages 905–918, Wiley-ISTE, London, 2008.
5. I. Aavatsmark, G. T. Eigestad, B. T. Mallison, J. M. Nordbotten: A compact
multipoint flux approximation method with improved robustness, Numerical Methods
for Partial Differential Equations, 24:1329–1360, 2008.
6. E. Keilegavlen, I. Aavatsmark: Monotonicity for control volume methods on
unstructured grids, In: 11th European Conference on the Mathematics of Oil Recovery,
EAGE, Bergen, 2008.
7. S. S. Mundal, E. Keilegavlen, I. Aavatsmark: Discretization schemes for anisotropic
heterogeneous problems on near-well grids, In: 11th European Conference on the
Mathematics of Oil Recovery, EAGE, Bergen, 2008.

Lectures 2008

1. Comparison of monotonicity for some multipoint flux approximation methods, 5th
International Symposium on Finite Volumes for Complex Applications, Aussois,
France, 13 June 2008.
2. Discretization principles of MPFA methods. Gordon Research Conference on Flow
and Transport in Permeable Media, Oxford, 16 July 2008.
3. Control-volume methods on non-K-orthogonal grids: Limitations and shortcomings,
European Consortium on Mathematics in Industry, London, 1 July 2008.
4. Anwendung von Dreiecken in Mehrpunktflussdiskretisierung. Kolloquium über
Angewandte Mathematik, Universität Erlangen, Erlangen, 30 September 2008.