Visualization of grids conforming to geological structures: a topological approach

Abstract

Flexible grids are used in many Geoscience applications because they can accurately adapt to the great diversity of shapes encountered in nature. These grids raise a number difficult challenges, in particular for fast volume visualization. We propose a generic incremental slicing algorithm for versatile visualization of unstructured grids, these being constituted of arbitrary convex cells. The tradeoff between the complexity of the grid and the efficiency of the method is addressed by special-purpose data structures and customizations. A general structure based on oriented edges is defined to address the general case. When only a limited number of polyhedron types is present in the grid (zoo grids), memory usage and rendering time are reduced by using a catalog of cell types generated automatically. This data structure is further optimized to deal with stratigraphic grids made of hexahedral cells. The visualization method is applied to several gridded subsurface models conforming to geological structures.

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:

• The geometry consists of a set of n vertices $\{v_1,\dots ,v_n\}$, each defined by a coordinate vector $\mathbf{x}=(x,y,z)$ in 3D space.

• 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.

• 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 $\phi$ (e.g., pressure, porosity, concentration) attached to the vertices.

There are several of challenges associated with unstructured grids:

• 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).

• 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).

• 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).

• 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 $\phi$ 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).