Coupling of MFE or Mimetic Finite Differences with Discontinuous Galerkin for Poroelasticity

1. Coupling of MFE or Mimetic Finite Differences with Discontinuous Galerkin for Poroelasticity Mary F. Wheeler Ruijie Liu Phillip Phillips Center for Subsurface Modeling The University of Texas at Austin

2. Vertical Subsidence Vertical Subsidence due to 100 million barrels of fluid (and sand) extracted from the Goose Creek oil field near Galveston, Texas (Pratt and Johnson, 1926, p.582). Water-covered areas are shown in black.

3. Bone Poroelasticity Cross-section of a long bone (Fritton, Wang, Weinbuam, and Cowin, 2001 Bioengineering Conference, ASME 2001). Remark: very low permeability

4. Poroelasticity Theory Governing Equations: Constitutive Laws:

5. Poroelasticity Theory Boundary conditions: Initial Conditions:

6. Notations, Spaces and Norms for Nonconforming Spaces

7. MFE/Mimetic Galerkin Formulation for Poroelasticity Discontinuous space of piecewise polynomials: Find such that: (1) (2) (3) where

8. Bilinear Form for Elasticity Bilinear Form: where Ref.: Riviere and Wheeler; Hansbo and Larson; Liu, Phillips and Wheeler, …

9. Auxiliary and Projection Error

10. References for Approximation Assumptions (Girault and Scott, 2000)

11. Error Estimates Main Results:

12. Error Estimates Main Results (Continued): Applying Gronwall's inequality: Displacement Flow and pressure (s : optimal exponent for flow)

13. Summary • If then the coupled model with mixed or mimetic finite elements for pressure and conforming Galerkin converges with optimal rates in energy and in L2 for flow pressure and velocity. Estimates depend on C*. Flow is locally conservative. • If discontinuous Galerkin is used and the approximation assumption holds, then couple mixed/mimetic or DG for flow and DG for displacements converge independent of C0(x). Flow is locally conservative.

14. Numerical Example Y Pressure Output Flag P0=1 psi X Red Line: No flow boundary Fixed displacement boundary P0=1 E: 1.0e+4 psi K=1.0e-6, 0.1 Kw =1.0e+12 time

15. Numerical Example DG: Solid is solved by discontinuous Galerkin finite element Mixed finite element for flow: piece-wise constant for pressure Flow DG for Solid

16. CG for Solid and MFE for Flow High Permeability

17. CG for Solid and MFE for Flow Low Permeability

18. CG for Solid and MFE for Flow DG: Low Permeability at earlier stage

19. Pressure Contour CG: Low Permeability DG: Low Permeability

20. 2F Y 2b X 2F 2a Mandel Problem

21. Analytical Pressure Solution of Mandel Problem

22. Numerical Results (CG for Solid and Flow) Incompressible Case Compressible Case CG: Linear-Linear; Low Permeability

23. Numerical Results (DG for Both Solid and Flow) Linear Elements Red line: CG Green Line: DG

24. Z pressure: 0.006 Mpa 10 mm 10 mm Y 5 mm 50 mm Width in X direction is 1 mm; E = 55 Mpa; Poisson's ratio =0.3 or 0.499; Uniform pressure = 0.006 Mpa Numerical Example—Bracket

25. Continuous Galerkin (a) Poisson Ratio = 0.3 (b) Poisson Ratio = 0.499

26. Discontinuous Galerkin: 0.499 (a) OBB (b) NIPG (c) SIPG (d) IIPG

27. Pure Bending Beam—CG and DG Simulations High Strength Materials Low Strength Material with Ideal Plasticity (Von-Mises )

28. Pure Bending Beam—Meshing Area where DG is applied

29. CG Simulation on Plastic Zone Development

30. CG Simulation on Plastic Zone Development

31. DG Simulation on Plastic Zone Development

32. Breast Reconstruction Model Continuous Galerkin Finite Element Solution • Elasticity Model • Large Deformation • Updated Geometry • Materials are close to incompressible • Poisson ratio = 0.499 • Gravity loading only • Domain in red is in tension

33. Breast Reconstruction Model Discontinuous Galerkin Finite Element Solution Poisson ratio = 0.499 Gravity loading only Domain in red is in tension

34. Breast Reconstruction Model Geometry updating for continuous gravity loading

35. Current and Future Work • Coupling of DG and CG in Geomechanics/Multiphase simulator • Extensions to include plasticity (Valhall Oil Reservoir) • Error estimators for adaptivity