Problem Definition: Solution of PDE’s in Geosciences

2. Problem Definition: Solution of PDE’s in Geosciences • Finite elements and finite volume require: • 3D geometrical model • Geological attributes and • Numerical meshes

3. Model Creation • 3D objects are defined by polygonal faces • Polygonal surfaces are input and intersected • A spatial subdivision is created • We require only the topological consistency of the input polygons • Vertices, edges and faces are constrained for meshing (internal and external boundaries)

4. Attributes • Horizons and faults are the building blocks • They have attributes, such as age and type • Attributes supply boundary conditions for PDE’s • The setting of attributes is not a simple task • Each vertex, edge, face has to know their horizons • A set of regions may correspond to a single layer

5. How to Generate Layers Automatically? • A 2.5D fence diagram • Two faults • Seven horizons

6. A Block Depicting Five Layers • Generally a layer is defined by two horizons, the eldest being at the bottom • Salt may cut several layers

7. The Algorithm • All regions have inward normals • We use the visibility of horizons from an outside point • The top horizon defines the layer • It has a negative volume and the greatest magnitude

8. A 3D Model With Four Layers • The blue layer is a salt diapir • All layers have been detected automatically

9. Automatic Mesh Generation • Three main families of algorithms • Octree methods • Delaunay based methods • Advancing front methods

10. Simple criteria for creating tetrahedra Unconstrained Delaunay triangulation requires only two predicates Point-in-sphere testing Point classification according to a plane Delaunay Advantages

11. No remarkable property in 3D Does not maximize the minimum angle as in 2D Constraining edges and faces may not be present (must be recovered later) May produce “useless” numerical meshes Slivers (“flat” tetrahedra) must be removed Delaunay Disadvantages

12. Background Meshes • The Delaunay criterion just tells how to connect points - it does not create new points • We use background meshes to generate points into the model • Based on crystal lattices • 20% of tetrahedra are perfect, even using the Delaunay criteria

13. Bravais Lattices • Hexagonal and Cubic-F (diamond) generate perfect tetrahedra in the nature

14. Size of a 3D triangulation Each vertex may generate in average 7 tets Multi-domain meshing Implies that each simplex has to be classified Mesh quality improvement Resulting mesh has to be useful in simulations Remeshing with deformation If the problem evolve over the time, the mesh has to be rebuilt as long as topology change Robustness Geological scale Challenges

15. Robustness • Automatic mesh generation requires robust algorithms • Robustness depends on the nature of the geometrical operations • We have robust predicates using exact arithmetic • Intersections cause robustness problems • Necessary to recover missing edges and faces • When applied to slivers may lead to an erroneous topology

16. The scale may vary from hundred of kilometers in X and Y To just a few hundred meters in Z Geological Scale

17. Non-uniform Scale • Implies bad tetrahedra shape. The alternative is either to: • Insert a very large number of points into the model, or • Refine the mesh, or • Accept a ratio of at least 10 to1

18. Multi-domain Models • We have to triangulate multi-domain models • Composed of several 3D internal regions • One external region • We have to specify the simplices corresponding to surfaces defining boundary conditions • This is necessary in finite element applications

19. A 45 Degree Cut of the Gulf of Mexico • 7 horizons • Bathymetri • Neogene • Paleogene • Upper Cretaceous • Lower Cretaceous • Jurassic • Basement

20. Cross Section of the Gulf of Mexico • Numbers • 2706 triangles • 4215 edges • 1210 vertices

21. Faces, edges and vertices on the boundary of the model are marked A point-in-region testing is performed for a single tetrahedron (seed) All tetrahedra reached from the seed without crossing the boundary are in the same region tetrahedra in the external region are deleted Simplex Classification

22. Gulf of Mexico Basin • Numbers • 6 regions • 63704 faces • 95175 edges • 31431 vertices

23. Triangulation of a Single Region • Numbers • 146373 tetrahedra • 1173 points automatically inserted • DA: [0.001241, 179.9] • Sa: [0.0, 359.2] • 2715 (1.854%) tets with min DA < 3.55 • 2257 out of 2715 tets with 4 vertices on constrained faces

24. Detail Showing Small Dihedral Angles

25. Conclusions • The use of a real 3D model opens a new dimension • Permits a much better understanding of geological processes • Multi-domain models are created by intersecting input surfaces • Must handle vertices closely clustered • Vertices in the range [10-7, 10+4] are not uncommon

26. Breaking the Egg • The ability of slicing a model reveals its internal structure.

27. Conclusions • Generation of 3D unconstrained Delaunay triangulation is straightforward • Hint: use an exact arithmetic package • The complicated part is to recover missing constrained edges and faces • Attributes must be present in the final mesh • We have a coupling during the mesh generation with the model being triangulated

28. Conclusions • The size of a tetrahedral mesh can be quite large • For a moderate size problem a laptop is enough