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Problem Definition Solution of PDEs in

Geosciences

- Finite elements and finite volume require
- 3D geometrical model
- Geological attributes and
- Numerical meshes

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)

Attributes

- Horizons and faults are the building blocks
- They have attributes, such as age and type
- Attributes supply boundary conditions for PDEs
- 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

How to Generate Layers Automatically?

- A 2.5D fence diagram
- Two faults
- Seven horizons

A Block Depicting Five Layers

- Generally a layer is defined by two horizons, the

eldest being at the bottom - Salt may cut several layers

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

A 3D Model With Four Layers

- The blue layer is a salt diapir
- All layers have been detected automatically

Automatic Mesh Generation

- Three main families of algorithms
- Octree methods
- Delaunay based methods
- Advancing front methods

Delaunay Advantages

- Simple criteria for creating tetrahedra

- Unconstrained Delaunay triangulation requires

only two predicates - Point-in-sphere testing
- Point classification according to a plane

Delaunay Disadvantages

- 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

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

Bravais Lattices

- Hexagonal and Cubic-F (diamond) generate perfect

tetrahedra in the nature

Challenges

- 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

- Size of a 3D triangulation
- Each vertex may generate in average 7 tets
- Multi-domain meshing
- Implies that each simplex has to be classified

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

Geological Scale

- The scale may vary from hundred of kilometers in

X and Y

- To just a few hundred meters in Z

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

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

A 45 Degree Cut of the Gulf of Mexico

- 7 horizons
- Bathymetri
- Neogene
- Paleogene
- Upper Cretaceous
- Lower Cretaceous
- Jurassic
- Basement

Cross Section of the Gulf of Mexico

- Numbers
- 2706 triangles
- 4215 edges
- 1210 vertices

Simplex Classification

- 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

- Faces, edges and vertices on the boundary of the

model are marked

Gulf of Mexico Basin

- Numbers
- 6 regions
- 63704 faces
- 95175 edges
- 31431 vertices

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 lt 3.55
- 2257 out of 2715 tets with 4 vertices on

constrained faces

Detail Showing Small Dihedral Angles

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, 104 are not

uncommon

Breaking the Egg

- The ability of slicing a model reveals its

internal structure.

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

Conclusions

- The size of a tetrahedral mesh can be quite large
- For a moderate size problem a laptop is enough

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