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

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

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

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

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

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

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:

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, …

C enter for S ubsurface M odeling Auxiliary and Projection Error

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

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

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

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

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

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

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

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

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

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

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

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

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

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

C enter for S ubsurface M odeling 50 mm Z pressure: 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 = Mpa Numerical Example—Bracket

C enter for S ubsurface M odeling (a) Poisson Ratio = 0.3 Continuous Galerkin (b) Poisson Ratio = 0.499

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

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 )

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

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

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

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

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 = Gravity loading only Domain in red is in tension Continuous Galerkin Finite Element Solution

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

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

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