Unstructured Voronoi grids conforming to lower-dimensional objects

Abstract

We present a novel mixed-dimensional method for generating unstructured polyhedral grids that conform to prescribed geometric objects in arbitrary dimensions. Two types of conformity are introduced: (i) control-point alignment of cell centroids to accurately represent horizontal and multilateral wells or create volumetric representations of fracture networks, and (ii) boundary alignment of cell faces to accurately preserve lower-dimensional geological objects such as layers, fractures, faults, and/or pinchouts. The prescribed objects are in this case assumed to be lower-dimensional, and we create a grid hierarchy in which each lower-dimensional object is associated with a lower-dimensional grid. Further, the intersection of two objects is associated with a grid one dimension lower than the objects. Each grid is generated as a clipped Voronoi diagram, also called a perpendicular bisector (PEBI) grid, for a carefully chosen set of generating points. Moreover, each grid is generated in such a way that the cell faces of a higher-dimensional grid conform to the cells of all lower-dimensional grids. We also introduce a sufficient and necessary condition which makes it easy to check if the sites for a given perpendicular bisector grid will conform to the set of prescribed geometric objects.