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.