Domain Decomposition Techniques to Couple Elasticity and Plasticity in Geomechanics

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Domain Decomposition Techniques to Couple
Elasticity and Plasticity in Geomechanics
By Horacio Florez, M.Sc., M.S.E.
Research work carried out under the supervision
of Dr. Mary F. Wheeler The Center for
Subsurface Modeling The University of Texas at
Austin
CSM Industrial Affiliates Meeting, Austin,
October 14, 2009
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Outline

• Summary and motivation
• Model problems
• Elasticity and loose coupling with flow ()
• Computational plasticity
• State-of-the-art Domain Decomposition schemes in
  geomechanics
• Dirichlet-Neumann ()
• Mortar FEM
• Numerical experiments in geomechanics on
  multi-core processors ()
• Concluding remarks, future work and references
• Part of summer internship work at The
  ConocoPhillips Company ()

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Summary
Were going to discuss popular Domain
Decomposition schemes to couple elasticity and
plasticity in geomechanics. The algorithm of
computational plasticity with popular failure
criteria will be discussed. We present the
computer implementation with FEM and preliminary
2-D results.
Top The Druker-Prager and Von Mises yield
surfaces are depicted. Left A plasticity front
starts to propagate due to the fact that the
material is yielding right there.
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Motivation Domain Decomposition
Elasticity and plasticity in geomechanics
iterative coupling
Reservoir level
Near borehole
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Outline

• Summary and motivation
• Model problems
• Elasticity and loose coupling with flow ()
• Computational plasticity
• State-of-the-art Domain Decomposition schemes in
  geomechanics
• Dirichlet-Neumann ()
• Mortar FEM
• Numerical experiments in geomechanics on
  multi-core processors ()
• Concluding remarks, future work and references
• Part of summer internship work at The
  ConocoPhillips Company ()

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Isotropic Elasticity

• FEM formulation

Fig. Applications include borehole stability and
sand production, reservoir compaction and
subsidence (loose coupling), among others
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Loose Coupling Reservoir Compaction

• Small perturbation assumption

• 1st problem at initial equilibrium, conditions
  are often unknown
• 2nd problem the stress changes due to pressure
  drop and changes in tractions no changes in body
  forces
• Superposition principle solve 2nd problem
  without knowing the 1st one

() Part of these ideas come from a geomechanics
course under Dr. Yves Leroy
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Computational Plasticity Algorithm

• Rate-independent plasticity (small deformation)

• Path-dependent materials

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Computational Plasticity Cont.

• The incremental boundary value problem

• Non-linear solution The Newton-Raphson scheme

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Plasticity Return Mapping Algorithm

• Fully implicit elastic predictor/return-mapping

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Outline

• Summary and motivation
• Model problems
• Elasticity and loose coupling with flow ()
• Computational plasticity
• State-of-the-art Domain Decomposition schemes in
  geomechanics
• Dirichlet-Neumann ()
• Mortar FEM
• Numerical experiments in geomechanics on
  multi-core processors ()
• Concluding remarks, future work and references
• Part of summer internship work at The
  ConocoPhillips Company ()

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Domain Decomposition
Dirichlet-Neumann (DN)

• Matching grids
• Coloring algorithm (3 colors tool)
• Over-relaxation is important
• Iterative coupling by the BCs

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DN Coloring and Algorithm

• White guys (D-guys) go first
• Hybrid grey guys proceed after D-guys
• Black ones can now go
• Feedback displacements to white and grey ones and
  go to step 1 if there is a residual in the
  tractions, stop if not

?mn
Fig. General partitioning will require a
three-color tool, hybrid sub-domains show up for
touching both D- and N- guys
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Speedup for Isotropic Elasticity
Kirschs Benchmark Problem

• Quadrilatheral mesh, nn 13564, ne 13280,
  Intel Xeon Processor E5440, 2.83 GHz, Quad Core
  (Harpertown)

() Part of internship work at The ConocoPhillips
Company
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Mortar FEM Method Elasticity

• Nonconforming discretizations

Fig. Non-matching interfaces and hanging-nodes
are treated properly
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Outline

• Summary and motivation
• Model problems
• Elasticity and loose coupling with flow ()
• Computational plasticity
• State-of-the-art Domain Decomposition schemes in
  geomechanics
• Dirichlet-Neumann ()
• Mortar FEM
• Numerical experiments in geomechanics on
  multi-core processors ()
• Concluding remarks, future work and references
• Part of summer internship work at The
  ConocoPhillips Company ()

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Reservoir Cross-Section Plane Strain
Boundary conditions and conforming mesh
Fig. The pressure field comes from a 100 x 20
black-oil model, the tensor product mesh
propagated in the surroundings is quite
inefficient and requires a non-matching treatment
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FEM Solution Conforming Mesh Case
Vertical displacement contour
Fig. The FEM solution shows compaction (in blue)
and build-up (in red)
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Reservoir Cross-Section Mortar Case
Boundary conditions and non-conforming mesh
Fig. The same tensor-product mesh is used in the
pay-zone while the surroundings are meshed with
Delaunay triangulations. The goal is to reduce
the computational cost
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FEM Solution with 4 Mortars
Vertical displacement contour
Fig. The mortar solution reproduces the same
features in the displacement field but the
computational cost was reduced by 50 because of
the efficient meshing
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Strip-Footing Plasticity Example
?
This problem allows determining the bearing
capacity (limit load) of a strip footing before
collapsing
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Strip-Footing FEM Solution
Vertical displacement contour and plasticity front
Fig. 1 The elastic trial (top) and the plastic
converged (bottom) solutions are shown for a
given load increment
Fig. 2 The plasticity front propagates during the
incremental loading process
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Outline

• Summary and motivation
• Model problems
• Elasticity and loose coupling with flow ()
• Computational plasticity
• State-of-the-art Domain Decomposition schemes in
  geomechanics
• Dirichlet-Neumann ()
• Mortar FEM
• Numerical experiments in geomechanics on
  multi-core processors ()
• Concluding remarks, future work and references
• Part of summer internship work at The
  ConocoPhillips Company ()

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Concluding Remarks

• We have presented
• Parallel Finite Element CG-Code was developed and
  tested on benchmark problems
• Domain Decomposition techniques for coupling
  elasticity and plasticity with DN and mortars
• Scalable speedup obtained for elasticity on 8
  processors with DN
• Scalable speedup achieved for plasticity up to 4
  cores (multi-threaded ensemble of tangent matrix)

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Future Work

• Further testing on Linux cluster machines like
  Bevo, Lonestar, and Ranger
• Implement other popular failure criteria such as
  Druker-Prager and Cam-Clay
• Benchmarking with both research and commercials
  codes such as HYPLAS, FEAP, Abaqus, etc.
• Incorporate more physics into the FEM-code
  thermal stresses and coupling with the energy
  equation
• We have to try with both Discontinuous Galerkin
  (DG)

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References Domain Decomp.

• Toselli, A. and Widlund, O., 2005, Domain
  Decomposition Methods Algorithms and Theory,
  Springer Series in computational Mathematics, New
  York, USA.
• Quarteroni, A. and Valli A., 1999, Domain
  Decomposition Methods for Partial Differential
  Equations, Numerical Mathematics and Scientific
  Computation , Oxford University Press, New York,
  USA.
• Girault, V., Pencheva, G., Wheeler, M. and,
  Wildey, T., 2009, Domain decomposition for
  linear elasticity with DG jumps and mortars,
  Comput. Methods Appl. Mech. Engrg., 198 (2009)
  1751-1765.
• Girault, V., Pencheva, G., Wheeler, M. and,
  Wildey, T., 2009, Domain decomposition for
  poro-elasticity with DG jumps and mortars, in
  preparation.
• Badia S. et al, 2009, Robin-Robin preconditioned
  Krylov methods for fluid-structure interaction
  problems, Comput. Methods Appl. Mech. Engrg.,
  198 (2009) 2768-2784.
• Discacciati M., et al., 2001, ROBIN-ROBIN DOMAIN
  DECOMPOSITION FOR THE STOKES-DARCY COUPLING,
  SIAM J. NUMER. ANAL., Vol. 45, No. 3, pp.
  1246-1268.
• Hauret, P. and Le Tallec, P., 2007, A
  discontinuous stabilized mortar method for
  general 3D elastic problems, Comput. Methods
  Appl. Mech. Engrg., 196 (2007) 4881-4900.
• Flemisch B., Wohlmuth, B. I., et al., 2005, A
  new dual mortar method for curved interfaces 2D
  elasticity, Int. J. Numer. Meth. Engng. 2005,
  68813-832.
• Hauret, P. and Ortiz, M., 2005, BV estimates for
  mortar methods in linear elasticity, Comput.
  Methods Appl. Mech. Engrg., 195 (2006)
  4783-4793.

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References Plasticity

• Neto, E. A. et al, 2008, Computational methods
  for plasticity theory and applications, Wiley,
  UK.
• Simo, J. C. and Hughes T.J.R., 1998,
  Computational Inelasticity, Springer,
  Interdisciplinary Applied Mathematics.
• Lubliner, J., 1990, Plasticity Theory , Dover
  Publications, Inc., New York.
• Zienkiewicz, O. C. and Cormeau, I.C., 1974,
  VISCO-PLASTICITY AND CREEP IN ELASTIC SOLIDS-
  UNIFIED NUMERICAL SOLUTION APPROACH,
  International Journal of Numerical Methods in
  Engineering , Vol. 8, pp. 821-845.
• Cormeau, I.C., 1975, NUMERICAL STABILITY IN
  QUASI-STATIC ELASTO/ VISCO-PLASTICITY,
  International Journal of Numerical Methods in
  Engineering , Vol. 9, pp. 109-127.
• Hughes, T.J.R. and Taylor, R. L., 1978,
  UNCONDITIONALLY STABLE ALGORITHMS FOR
  QUASI-STATIC ELASTO/ VISCO-PLASTIC FINITE
  ELEMENT ANALYSIS, Computers Structures, Vol.
  8, pp. 169-173.
• Simo, J. C. and Taylor, R. L., 1985, CONSISTENT
  TANGENT OPERATORS FOR RATE INDEPENDENT
  ELASTOPLASTICITY, Computer Methods in Applied
  Mechanics and Engineering, Vol. 48, pp. 101-118.
• Simo, J. C. and Taylor, R. L., 1986, A RETURN
  MAPPING ALGORITHM FOR PLANE STRESS
  ELASTOPLASTICITY, International Journal of
  Numerical Methods in Engineering, Vol. 22, pp.
  649-670.
• Wilkins, M.L., 1964, Calculation of
  Elasto-Plastic Flow, In Methods of Computational
  Physics 3, eds. , B. Alder et. al., Academic
  Press, New York.
• Clausen, J., et al., 2007, An efficient return
  mapping algorithm for non-associated plasticity
  with linear yield criteria in principal stress
  plane, Computers Structures, Vol. 85, pp.
  1975-1807.

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References Poroelasticity

• Kim, J. et al., 2009, Stability, Accuracy and
  Efficiency of Sequential Methods for Coupled Flow
  and Geomechanics, SPE Paper 119084.
• Liu R., 2004, Discontinuous Galerkin Finite
  Element Solution for Poromechanics, PhD thesis,
  The University of Texas at Austin .
• Gai X., 2004, A Coupled Geomechanics and
  Reservoir Flow Model on Parallel Computers, PhD
  thesis, The University of Texas at Austin .
• Han G. et al., 2002, Semi-Analytical Solutions
  for the Effect of Well Shut Down on Rock
  Stability, Canadian International Petroleum
  Conference, Calgary, Alberta .
• Chen Z, et al, 2006, Computational Methods for
  Multiphase Flows in Porous Media, SIAM, pp. 57
  247-258 .
• Du J, and Olson J., 2001, A poroelastic
  reservoir model for predicting subsidence and
  mapping subsurface pressure fronts, Journal of
  Petroleum Technology Science, Vol. 30, pp.
  181-197.
• Grandi, S. and Nafi M., 2001, Geomechanical
  Modeling of In-situ Stresses around a Borehole,
  MIT, Cambridge, MA.
• Charlez A., 1999, The concept of Mud Window
  Applied to Complex Drilling, SPE Paper 56758 .

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Contact Us
Visit us http//www.ices.utexas.edu/subsurface
/ e-Mail florezg_at_gmail.com Any
Questions?
End of presentation
Thanks for your attention
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Rate Independent Plasticity
We just follow the approach by Simo and Hughes
(1998) and Lubliner (1990)
Elastic domain and yield criterion
Flow rule and hardening law
Kuhn-Tucker complementary conditions
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Rate Independent Plasticity
Interpretation of the Kuhn-Tucker complementary
conditions
Consistency condition and elastoplastic tangent
moduli
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Rate Independent Plasticity
Assumption for the flow rule, hardening law, and
yield condition satisfy
Finally the so called tensor of tangent
elastoplastic moduli becomes
For the special case of associative flow rule we
have
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Failure Criteria
We just follow the approach by Zienkiewicz and
Cormeau (1974) and Hughes (1978)
The visco-plastic strain rate law
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Druker-Prager Yield Surface
Common expressions are given by