Visualization of grids conforming to geological structures: a topological approach☆
Introduction
Most partial differential equations (PDEs) describing natural processes in the subsurface are too complex to be solved analytically. The solution to these equations is usually approximated by a numerical method (e.g., finite differences), hence relies on the definition of a grid discretizing the domain of study. This grid may be Cartesian, but geological heterogeneities are modeled more accurately and at a lower computational cost using a flexible grid made to conform to sedimentary and tectonic structures. For flow simulation, a flexible grid can be further optimized to account for radial flow around wells (Heinemann et al., 1991, Aziz, 1993, Verma and Aziz, 1997, Mlacnik et al., 2003). Such a flexible grid may also be used in Geographical Information Systems (Breunig, 1999) and in model building (Courrioux et al., 2001, Hale, 2002, Mallet, 2004). A grid is generally defined by three components:
- 1.
The geometry consists of a set of n vertices , each defined by a coordinate vector in 3D space.
- 2.
The topological model describes how these vertices are connected to each other (Fig. 1), defining a partition of the space into polyhedral 3-cells (or, more simply, cells); each cell c is defined by its faces, each face by its edges, and each edge by its bounding vertices. As opposed to a structured grid in which connection patterns are repeated as in a crystal lattice, an unstructured grid must have its topology explicitly defined. A zoo grid is an unstructured grid containing only a small number of cell types.
- 3.
The property model assigns property values to the vertices, the edges, the faces or the cells of the grid. For the sake of clarity, this paper describes only the case of one scalar property value (e.g., pressure, porosity, concentration) attached to the vertices.
- 1.
The gridding methods must find a compromise between conflicting criteria such as conformity to geological structures vs. cell numbers, sizes and shapes (e.g., Owen, 1998, Lepage, 2002).
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As a counterpart for better accuracy, the discretization of PDEs is more difficult to define on unstructured grids than on Cartesian grids (Verma and Aziz, 1997).
- 3.
Existing implementations of geostatistical algorithms cannot be used directly; new implementations for unstructured grids rely on efficient neighborhood search algorithms, and must account for variable cell volumes (Deutsch et al., 2002).
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Most volume visualization techniques, essential for quality control and perception of geological volumes, were also developed on Cartesian grids, hence must be adapted to unstructured grids (e.g., Shirley and Tuchman, 1990, Williams et al., 1998, Silva and Mitchell, 1997, Lévy et al., 2001).
This paper is concerned with this last point, namely the volume visualization of a scalar field defined on an unstructured grid. Producing explicit images from a volume grid is always challenging, for a large amount of data has to be traversed and organized in real time. The way a grid is structured conditions the applicability and the efficiency of a particular rendering algorithm. Whereas most existing methods practically handle only tetrahedral grids (Shirley and Tuchman, 1990, Silva and Mitchell, 1997, Cignoni et al., 1998, Wittenbrink, 1999), we consider the more general case where grid cells are arbitrary convex polyhedra.
After a short review of existing volume visualization techniques (Section 2), we present a new imaging algorithm, accepting any grid composed of convex cells (Sections 3 and 4). This algorithm is then specialized for zoo grids for increased memory and time efficiency (Section 5).
Section snippets
Visualization techniques
Two types of approaches have been defined to visualize the interior of a volume grid (e.g., Kaufman, 1996):
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Surface-based techniques use classical 3D drawing of a surface extracted from the grid. Such a surface may be a cross-section or, more generally, an equipotential of some scalar field defined on the grid (isosurface rendering). Note that an arbitrary planar section is just a particular isosurface where , with being real coefficients and the spatial
Principle
We propose to compute a set of isosurfaces between two values and separated by a interval.
On a grid made of cells and rendered with slices, a naive algorithm trying to check each cell against every isosurface would have an average complexity, whereas only cells are intersected by a slice.
In this respect, the contour seeds method (Bajaj et al., 1996) is probably optimal: for each connected component of the isosurface, all the intersected cells are found
Circular incident edge lists (CIEL)
This section is a revisit of our previous work (Lévy et al., 2001). The memory requirements of the data structure have been reduced without loss of performance by eliminating the use of an edge structure. The active edges update has been improved, using two FIFO queues in place of a list of active edges, and the complete description of the construction algorithm is given.
Cellular graphs
The CIEL implementation of our generic volume rendering algorithm handles arbitrary polyhedral grids made of convex cells. Yet, such a generality has a cost in terms of memory requirements and performance. In this section, we address the simpler class of zoo grids (Fig. 1). This type of grid can be generated by mixing structured parts with hexahedral cells and unstructured parts with tetrahedra and pyramids, e.g., at the neighborhood of fractures.
Results
The visualization method described here has been implemented in using generic programming. It was applied to several grids of various topologies (Table 1, Fig. 9): the Voronoi data set is a synthetic grid created from the Delaunay tetrahedralization of a point cloud, as encountered in reservoir simulation applications. The SGrid data set is a faulted stratigraphic grid made of hexahedral cells, and the Warped Skull is a warped medical grid.
For each of these data sets, storage cost is
Conclusion
Based on a general efficient incremental slicing algorithm, we have presented a generic method for visualizing unstructured grids with convex cells through volume rendering or isosurface extraction. This method has been applied to the various types of grids encountered in Geosciences, namely polyhedral grids (CIEL), zoo and homogeneous grids (CGraphs), and curvilinear stratigraphic grids (structured CGraphs). This method generalizes previous approaches based on the sweeping paradigm (Silva and
Acknowledgements
This work was supported by the ACI Geogrid (French Ministry of Research), the ARC Plasma (INRIA), and ChevronTexaco. Part of this work was performed in the frame of the Gocad Consortium (http://gocad.ensg.inpl-nancy.fr). Thanks to Total for the SGrid data set, and to Siemens Medical Systems for the skull data set.
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Code available from http://www.loria.fr/~levy/Graphite