Forming a Triangulated Grid of a 3D Fracture Network

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Forming a Triangulated Grid of a 3D Fracture
Network

• Dr Michael Robins UWA CWR Sept 04

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3-Phase NAPL Transport Model
Ref D.Reynolds and B.Kueper 2001
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Fracture Networks
Three dimensions
Ref Reynolds Kueper 2003
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Cross section of 3D Fracture Network
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Representation of fractures as discs
a

• position (x,y,z)
• radius r
• orientation (u,v,w)
• aperture a

(u,v,w)
z
r
y
x
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Fracture network is formed by intersecting discs
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Triangulation of a set of vertices
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Delaunay Triangulation

• No vertices within circumcircle of triangles
• Many algorithms
• Incremental refinement
• Sweep line
• Divide and conquer
• Good for numerical models
• Constrained by Planar Straight Line Graphs
• Conform to angle or area restrictions

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Triangulation of Discs
Delaunay Triangulation
Conforming Delaunay angle and area limits
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Triangulation of intersecting discs
Constrained conforming Delaunay triangulation

• Planar Straight Line Graph
• Intersections must be included
• Steiner points may be added along line segments
• Added points must exist in both intersected discs

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Triangle data structures

• Vertex
• position
• Triangle
• vertices
• neighbours
• adjacent edges
• Edge
• colinear neighbours
• adjacent triangles
• Triangles and Edges
• handled or grasped relative to several
  orientations

Ref J. R. Shewchuk, 2002
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The Intersecting disc problem

• For a huge collection of seemingly
  randomly oriented, sized and positioned discs
• Form a triangulated network of vertices that lie
  on the disks and intersections
• Solution must suit parallel computation
• Novelty lies in the requirement that
  inserted vertices along the intersection
  (Steiner points) must be present at
  corresponding places in both discs

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2½ Dimension Triangulation

• A disc is a 12 sided polygon not a circle
• Triangulation problem is 2 dimensional
• Discs are flat therefore intersections are
  striaght
• Various overlaps but all intersect the boundary
  of one disc or the other
• Difficulty lies in identifying shared points of
  intersecting discs
• Work in 2D as much as possible

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Robust Arithmetic

• Calculations performed using floating point
• Limited numerical precision of the computer
• The solution must be unambiguous
• Problems are inconsistencies of data structures
  and triangles, eg negative areas etc.
• Shewchuks robust math routines perform
  calculations to increasing precision until result
  is unambiguous

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Discus program to triangulate intersecting discs

• Represent each disc as 2D vertices
• Shared vertices along intersections
• Represented as both 3D locations
• And corresponding points calculated in frame of
  reference of each disc
• Shared vertices are tagged with the same id.
• By using separate 2D representations, all the
  triangulation difficulties are already dealt with
  by Shewchuks algorithms

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How to calculate the intersections
foreach disc determine a bounding box sort all
discs by Lowest, Closest, Leftmost bounding
point Repeat with the first_disc remaining in
the sorted list read discs while disc.Lowest
ClosestLeft gt first_disc.HighestFarthestRight fo
reach disc read compare bounding boxes of
disc with first_disc foreach intersecting
boxes check if the discs actually
intersect write out first_disc and its
intersections until all discs have been examined
Avoid excessinve comparisions by keeping lists
of previously considered pairings
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Check if and where two specific discs intersect
calculate the line of intersection of the discs
planes from orientation norms and centre
points translate the line into the frame of
reference of each disc see if the line
intersects and boundary segments of the
disc this yields 4 points (2 or 3 in degenerate
cases) determine the overlap and the two shared
vertices or no intersection create the Planar
Straight Line Graph for each disc
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Additional considerations

• Three discs intersect at a point
• Additional shared vertex needs to be inserted
  into the PSLG of each disc
• Three discs intersect along a line and
  intersecting intersections may form acute angles
• Want to merge some shared vertices and realign
  structures slightly
• Tangential intersections
• Can be considered as no connection
• Boundary of the domain
• Post processing step to cut the triangulation

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Frame of reference for a disc

• Arbitrarily formed by
• Shifting the disc to the origin
• Rotating about the line z0
• Lie the disc flat in the xy plane
• Transform is unique for each disc
• Disc then formed in xy coordinates
• 12 vertices in 2D

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Forming a 3D unstructured grid of a fracture
network

• Transforming the private vertices of the discs
  into 3D
• Restore their connectivity
• Connect all the shared vertices (already in 3D)
  to respective nodes
• Note how the shared vertices provide the entire
  linkage of discs to each other

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Triangulation with shared verticesongoing work

• Delaunay Triangulation requires insertion of
  Steiner points along intersections.
• Any such insertion must be shared.
• The Delaunay and Constrained Delaunay algorithms
  tend to work by successive refinement
• Postulate that insertion of shared vertices can
  be coordinated between separate triangulation
  processes
• The algorithm would seem to suit parallel
  implementation
• natural and reasonably minimal communication
  patterns

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Tools

• C language triangle.c, robust.c and
  mytriangle.c
• Python a handy scripting language
• Generating or manipulating data files
• Prototype algorithm development
• Mayavi Python based visualization tool
• Vtk visualization toolkit
• Open source, used by Mayavi

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Conclusion

• Have determined the algorithm and equations
• Partially implemented transforms and
  triangulation
• Data structures lead to difficulties debugging
• Visualization tools are proving useful

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Bibliography

• David A. Reynolds and Bernard H. Kueper,
  Multiphase flow and transport in fractured clay /
  sand sequences, J. Contaminant Hydrology 51(2001)
  pp. 41-62, 2001
• David A. Reynolds and Bernard H. Kueper,
  Effective constitutive properties for dense
  nonaqueous phase liquid (DNAPL) migration in
  large fracture networks A computational study,
  Water Resources Research, 39(9), 2003
• Jonathan Richard Shewchuk, Delaunay Refinement
  Algorithms for Triangular Mesh Generation,
  Computational Geometry Theory and Applications
  22(1-3)21-74, May 2002.
• Marshall Bern and David Eppstein, Mesh Generation
  and Optimal Triangulation, pp. 23-90 of Computing
  in Euclidean Geometry, Ding-Zhu Du and Frank
  Hwang (editors), World Scientific, Singapore,
  1992.
• Jim Ruppert, A Delaunay Refinement Algorithm for
  Quality 2-Dimensional Mesh Generation, Journal of
  Algorithms 18(3)548-585, May 1995.
• L. Paul Chew, Guaranteed-Quality Mesh Generation
  for Curved Surfaces, Proceedings of the Ninth
  Annual Symposium on Computational Geometry (San
  Diego, California), pages 274-280, Association
  for Computing Machinery, May 1993.
• Marshall Bern, David Eppstein, and John R.
  Gilbert, Provably Good Mesh Generation, Journal
  of Computer and System Sciences 48(3)384-409,
  June 1994.
• Jonathan Richard Shewchuk, Adaptive Precision
  Floating-Point Arithmetic and Fast Robust
  Geometric Predicates, Discrete Computational
  Geometry 18305-363, 1997.