1. C enter for S ubsurface M odeling 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. C enter for S ubsurface M odeling 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. Vertical Subsidence

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

4. C enter for S ubsurface M odeling Governing Equations: Constitutive Laws: Poroelasticity Theory

5. C enter for S ubsurface M odeling Poroelasticity Theory Boundary conditions: Initial Conditions:

6. C enter for S ubsurface M odeling Notations, Spaces and Norms for Nonconforming Spaces

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

8. C enter for S ubsurface M odeling Bilinear Form: where Bilinear Form for Elasticity Ref.: Riviere and Wheeler; Hansbo and Larson; Liu, Phillips and Wheeler, …

9. C enter for S ubsurface M odeling Auxiliary and Projection Error

10. C enter for S ubsurface M odeling References for Approximation Assumptions (Girault and Scott, 2000)

11. C enter for S ubsurface M odeling Error Estimates Main Results:

12. C enter for S ubsurface M odeling Main Results (Continued): Applying Gronwall's inequality: DisplacementFlow and pressure ( s : optimal exponent for flow) Error Estimates

13. C enter for S ubsurface M odeling If then the coupled model with mixed or mimetic finite elements for pressure and conforming Galerkin converges with optimal rates in energy and in L 2 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 C 0 (x). Flow is locally conservative. Summary

14. C enter for S ubsurface M odeling P 0 =1 psi Fixed displacement boundary Red Line: No flow boundary E: 1.0e+4 psi K=1.0e-6, 0.1 Kw =1.0e+12 P 0 =1 time Pressure Output Flag X Y Numerical Example

15. C enter for S ubsurface M odeling DG: Solid is solved by discontinuous Galerkin finite element Mixed finite element for flow: piece-wise constant for pressure DG for Solid Flow Numerical Example

16. C enter for S ubsurface M odeling High Permeability CG for Solid and MFE for Flow

17. C enter for S ubsurface M odeling Low Permeability CG for Solid and MFE for Flow

18. C enter for S ubsurface M odeling DG: Low Permeability at earlier stage CG for Solid and MFE for Flow

19. C enter for S ubsurface M odeling CG: Low PermeabilityDG: Low Permeability Pressure Contour

20. C enter for S ubsurface M odeling Mandel Problem 2F 2a 2b X Y

21. C enter for S ubsurface M odeling Analytical Pressure Solution of Mandel Problem

22. C enter for S ubsurface M odeling CG: Linear-Linear; Low Permeability Incompressible CaseCompressible Case Numerical Results (CG for Solid and Flow)

23. C enter for S ubsurface M odeling Numerical Results (DG for Both Solid and Flow) Linear Elements Red line: CG Green Line: DG

24. C enter for S ubsurface M odeling 50 mm Z pressure: 0.006 Mpa 5 mm 10 mm Y 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. C enter for S ubsurface M odeling (a) Poisson Ratio = 0.3 Continuous Galerkin (b) Poisson Ratio = 0.499

26. C enter for S ubsurface M odeling (a) OBB Discontinuous Galerkin: 0.499 (b) NIPG (c) SIPG (d) IIPG

27. C enter for S ubsurface M odeling Pure Bending Beam—CG and DG Simulations High Strength Materials Low Strength Material with Ideal Plasticity (Von-Mises )

28. C enter for S ubsurface M odeling Pure Bending Beam—Meshing Area where DG is applied

29. C enter for S ubsurface M odeling CG Simulation on Plastic Zone Development

30. C enter for S ubsurface M odeling CG Simulation on Plastic Zone Development

31. C enter for S ubsurface M odeling DG Simulation on Plastic Zone Development

32. C enter for S ubsurface M odeling Breast Reconstruction Model 1.Elasticity Model 2.Large Deformation 3.Updated Geometry 4.Materials are close to incompressible Poisson ratio = 0.499 Gravity loading only Domain in red is in tension Continuous Galerkin Finite Element Solution

33. C enter for S ubsurface M odeling Breast Reconstruction Model Discontinuous Galerkin Finite Element Solution Poisson ratio = 0.499 Gravity loading only Domain in red is in tension

34. C enter for S ubsurface M odeling Breast Reconstruction Model Geometry updating for continuous gravity loading

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