Topological visualization of tensor fields using a generalized Helmholtz decomposition

Analysis and visualization of fluid flow datasets has become increasing important with the development of computer graphics. Even though many direct visualization methods have been applied in the tensor fields, those methods may result in much visual clutter. The Helmholtz decomposition has been widely used to analyze and visualize the vector fields, and it is also a useful application in the topological analysis of vector fields. However, there has been no previous work employing the Helmholtz decomposition of tensor fields. We present a method for computing the Helmholtz decomposition of tensor fields of arbitrary order and demonstrate its application. The Helmholtz decomposition can split a tensor field into divergence-free and curl-free parts. The curl-free part is irrotational, and it is useful to isolate the local maxima and minima of divergence (foci of sources and sinks) in the tensor field without interference from curl-based features. And divergence-free part is solenoidal, and it is useful to isolate centers of vortices in the tensor field. Topological visualization using this decomposition can classify critical points of two-dimensional tensor fields and critical lines of 3D tensor fields. Compared with several other methods, this approach is not dependent on computing eigenvectors, tensor invariants, or hyperstreamlines, but it can be computed by solving a sparse linear system of equations based on finite difference approximation operators. Our approach is an indirect visualization method, unlike the direct visualization which may result in the visual clutter. The topological analysis approach also generates a single separating contour to roughly partition the tensor field into irrotational and solenoidal regions. Our approach will make use of the 2nd order and the 4th order tensor fields. This approach can provide a concise representation of the global structure of the field, and provide intuitive and useful information about the structure of tensor fields. However, this method does not extract the exact locations of critical points and lines

3D Seismic Interpretation and Well Log Analysis of the Marcellus Shale of Appalachian Basin at Taylor County, West Virginia

The organic-rich Marcellus Shale unit of Middle Devonian section in Appalachian basin currently is one of the most successful unconventional shale-gas reservoirs in the world because of the modern technology development including; advanced seismic imaging, hydraulic fracturing and horizontal drilling. This study extracts the geologic structure and discontinuity systems from seismic data in Taylor County, West Virginia to better understand the regional and local geologic setting with special emphasis on the orientation of structural features (e.g., folds, faults and joints) that can guide horizontal well drilling and fracture stimulation design.;A southwest-northeast synclinal fold was mapped in the northern part of seismic survey area and a parallel partial anticline in the southern part of seismic survey area. There is a crossstrike structural discontinuity (CSD\u27s) crosses the northeast part of study area, which is orientating northwest-southeast and cuts from Cambrian-Knox Group to Upper Devonian-Elk Group. The thickness of Marcellus Shale interval across the study area is almost uniform (30 meters). There is only one seismically-visible reverse fault across the Marcellus Shale. But a large thrust fault crosses the Trenton Limestone, Reedsville Group and Juniata Group. Some detachment appears within the Rome Shale, and Beekmantown, Reedsville, Juniata and Elk groups. Detachment was not observed in the Salina Group. Discontinuity attributes were used to extract the discontinuity system from seismic data. Attributes used includes variance, chaos, dip deviation, 3D curvature and Ant tracking. The 3D curvature attribute images the discontinuity feature within the Hamilton Group, which matches the geologic structure contour lines of Hamilton Group. The Ant tracking attribute shows the orientation and distribution of discontinuities across the study area, which almost match the geologic structure contour lines of Hamilton Group. Furthermore, the Ant tracking of Marcellus Shale displays a fabric, which strongly indicates two regional joint sets, earlier J1 crosscut by later J2 sets. As measured from Ant tracking, the J1 set is orientated approximately N52E, and the J2 set is trending around N45W. There is a low angle between the J1 set and the maximum compressive normal stress of the contemporary tectonic stress field (S Hmax).;Similar to other parts of the Appalachian basin, horizontal drilling should be N38W and perpendicular to the J1 set to cross and utilize the joints to guide fracture stimulation. There are areas of crosscutting N45W discontinuities that are interpreted as fracture swarms, which is related to cross-strike structural discontinuity structures. These areas may form reservoir barriers or areas where hydraulic fracturing could be ineffective due to energy largely loss along faults and open fractures

Visualizing High-Order Symmetric Tensor Field Structure with Differential Operators

The challenge of tensor field visualization is to provide simple and comprehensible representations of data which vary both directionally and spatially. We explore the use of differential operators to extract features from tensor fields. These features can be used to generate skeleton representations of the data that accurately characterize the global field structure. Previously, vector field operators such as gradient, divergence, and curl have previously been used to visualize of flow fields. In this paper, we use generalizations of these operators to locate and classify tensor field degenerate points and to partition the field into regions of homogeneous behavior. We describe the implementation of our feature extraction and demonstrate our new techniques on synthetic data sets of order 2, 3 and 4