Accurate Discretizations on Complex Grids

GeoScale - Direct Reservoir Simulation on Geocellular Models

Basic problem

Accurate geological modelling of features such as faults, fractures, erosion and horizontal wells requires grids that are flexible with respect to geometry. The industry standard is to use so-called stratigraphic grids (corner-point, PEBI, etc) as a means to represent the reservoir stratigrapy. Such grids generally contain polyhedral cells and complex grid cell connectivities. The most common discretization method, the two-point flux-approximation (TPFA) method, is only consistent on K-orthogonal grids and may exhibit (severe) grid-orientation effects unless special care is take during gridding. Modern discretization methods like the multipoint flux-approximation (MPFA) methods and the mimetic method eliminate this problem. Our research has focused on adapting the mimetic method to grids with industry-standard complexity and have used the resulting method as a consistent and accurate building block for constructing multiscale methods. In particular, using mimetic methods eliminates grid-orientation effects and makes it possible to model fault geometry accurately without introducing discretization errors.

The mimetic method

The method is based on making an approximation to the discret inner product for the vector field. To achive a consistent and convergent method, this inner product has to reproduce linear flow exactly. A bit simplified, on may say that the method has flexibility of a finite-element method and the simplicity of a finite-difference method.

Advantages

convergent and consistent discretization
reduces grid-orientation effects significantly compared with the standard two-point discretization
has a generic implementation for grids consisting of polyhedral cells with arbitrary number of faces
results in a symetric positive-definite system
has (simple) extensions that can be used on curved surfaces.

References

K.-A. Lie, S. Krogstad, I. S. Ligaarden, J. R. Natvig, H. M. Nilsen, and B. Skaflestad. Open source MATLAB implementation of consistent discretisations on complex grids. Comput. Geosci.
H. M. Nilsen, K.-A. Lie, and J. R. Natvig. Accurate modelling of faults by multipoint, mimetic, and mixed methods. SPE J., to appear, DOI: 10.2118/149690-PA.
J. E. Aarnes, S. Krogstad, and K.-A. Lie. Multiscale mixed/mimetic methods on corner-point grids. Comput. Geosci., Vol. 12, No. 3, pp. 297-315, 2008. DOI: 10.1007/s10596-007-9072-8

Reproduction of linear flow for three different pressure solvers (pressure drop from left to right, homogeneous permeability with anisotropy ratio 1 : 1000 aligned with the grid). Upper row: velocity field, bottom row: pressure.

Analytical and numerical solutions for the flattened top layer of a real-field model; grid-orientation effects are easily seen for the TPFA method

Published June 6, 2011

A portfolio of strategic research projects funded by the Research Council of Norway