Cell Centered Finite Volume Schemes for Multiphase Flow Applications - PowerPoint PPT Presentation

Outline Applications and meshes Flux formulation Discrete variational framework Numerical Experiments Cell Centered Finite Volume Schemes for Multiphase Flow Applications L. Agelas 1 , D. Di Pietro 1 , J. Droniou 2 , I. Kapyrin 1 , R. Eymard 3 , C. Guichard 1 , R. Masson 1 . 1 Institut Français du Pétrole 2 Université de Montpellier 3 Université Paris Est june 22-24th 2009 L. Agelas1, D. Di Pietro1, J. Droniou2, I. Kapyrin1, R. Eymard3, C. Guichard1, R. Masson1. Cell Centered Finite Volume Schemes for Multiphase Flow Applications

Outline Applications and meshes Flux formulation Discrete variational framework Numerical Experiments Applications and meshes 1 Flux formulation 2 The L and G schemes Discrete variational framework 3 The GradCell scheme The O scheme Numerical Experiments 4 L. Agelas1, D. Di Pietro1, J. Droniou2, I. Kapyrin1, R. Eymard3, C. Guichard1, R. Masson1. Cell Centered Finite Volume Schemes for Multiphase Flow Applications

Outline Applications and meshes Flux formulation Discrete variational framework Numerical Experiments Applications Basin Modeling Reservoir simulation CO2 geological storage L. Agelas1, D. Di Pietro1, J. Droniou2, I. Kapyrin1, R. Eymard3, C. Guichard1, R. Masson1. Cell Centered Finite Volume Schemes for Multiphase Flow Applications

Outline Applications and meshes Flux formulation Discrete variational framework Numerical Experiments Meshes: corner point geometries with faults L. Agelas1, D. Di Pietro1, J. Droniou2, I. Kapyrin1, R. Eymard3, C. Guichard1, R. Masson1. Cell Centered Finite Volume Schemes for Multiphase Flow Applications

Outline Applications and meshes Flux formulation Discrete variational framework Numerical Experiments Meshes: corner point geometries with erosions L. Agelas1, D. Di Pietro1, J. Droniou2, I. Kapyrin1, R. Eymard3, C. Guichard1, R. Masson1. Cell Centered Finite Volume Schemes for Multiphase Flow Applications

Outline Applications and meshes Flux formulation Discrete variational framework Numerical Experiments Meshes: basin geometries L. Agelas1, D. Di Pietro1, J. Droniou2, I. Kapyrin1, R. Eymard3, C. Guichard1, R. Masson1. Cell Centered Finite Volume Schemes for Multiphase Flow Applications

Outline Applications and meshes Flux formulation Discrete variational framework Numerical Experiments Meshes: nearwell meshes L. Agelas1, D. Di Pietro1, J. Droniou2, I. Kapyrin1, R. Eymard3, C. Guichard1, R. Masson1. Cell Centered Finite Volume Schemes for Multiphase Flow Applications

Outline Applications and meshes Flux formulation Discrete variational framework Numerical Experiments Difficulties: Geometry Degenerated cells due to erosion Dynamic mesh (basin models): the scheme must be recomputed at each time step Faults in basin models: geometry not always available (overlaps and holes) Conductive Faults in basin models General polyhedral cells Submeshes (dead cells) Local Grid Refinement Adaptive Mesh Refinement Boundary conditions L. Agelas1, D. Di Pietro1, J. Droniou2, I. Kapyrin1, R. Eymard3, C. Guichard1, R. Masson1. Cell Centered Finite Volume Schemes for Multiphase Flow Applications

Outline Applications and meshes Flux formulation Discrete variational framework Numerical Experiments Difficulties: complex physics Heterogeneous anisotropic media Dispersion (full tensor,time and space dependent) Multiphase Darcy Flows Complex closure laws: thermodynamical equilibrium, geochemistry, Kinetics Thermics Geomechanics L. Agelas1, D. Di Pietro1, J. Droniou2, I. Kapyrin1, R. Eymard3, C. Guichard1, R. Masson1. Cell Centered Finite Volume Schemes for Multiphase Flow Applications

Outline Applications and meshes Flux formulation Discrete variational framework Numerical Experiments Motivations of cell centered schemes for compositional multiphase Darcy flow applications N c primary unknowns per cell for multiphase compositional flows Explicit linear fluxes Easier to combine TPFA and MPFA Existing Efficient Preconditioners like CPR-AMG Adapted to fully or semi implicit discretizations of multiphase compositional Darcy flows But “compact” MPFA VF schemes are non symmetric on general meshes Possible lack of robustness due to mesh and diffusion coefficients dependent coercivity (linear solver, pressure convergence) and monotonicity (non linear solver) L. Agelas1, D. Di Pietro1, J. Droniou2, I. Kapyrin1, R. Eymard3, C. Guichard1, R. Masson1. Cell Centered Finite Volume Schemes for Multiphase Flow Applications

Outline Applications and meshes Flux formulation Discrete variational framework Numerical Experiments ArcGeoSim platform Based on Arcane Platform co-developped by CEA-DAM and IFP L. Agelas1, D. Di Pietro1, J. Droniou2, I. Kapyrin1, R. Eymard3, C. Guichard1, R. Masson1. Cell Centered Finite Volume Schemes for Multiphase Flow Applications

Outline Applications and meshes Flux formulation Discrete variational framework Numerical Experiments Cell centered schemes currently implemented in ArcGeoSim L and G schemes O scheme GradCell scheme L. Agelas1, D. Di Pietro1, J. Droniou2, I. Kapyrin1, R. Eymard3, C. Guichard1, R. Masson1. Cell Centered Finite Volume Schemes for Multiphase Flow Applications

Outline Applications and meshes Flux formulation The L and G schemes Discrete variational framework Numerical Experiments Model problem Let Ω ⊂ R d be a bounded polygonal domain For f ∈ L 2 (Ω) , consider the following problem: � − div ( ν ∇ u ) = f in Ω , u = 0 on ∂ Ω � Let a ( u , v ) = Ω ν ∇ u · ∇ v . The weak formulation reads � Find u ∈ H 1 fv for all v ∈ H 1 0 (Ω) such that a ( u , v ) = 0 (Ω) ( Π ) Ω L. Agelas1, D. Di Pietro1, J. Droniou2, I. Kapyrin1, R. Eymard3, C. Guichard1, R. Masson1. Cell Centered Finite Volume Schemes for Multiphase Flow Applications

Outline Applications and meshes Flux formulation The L and G schemes Discrete variational framework Numerical Experiments Model problem Let { Ω i } 1 ≤ i ≤ N Ω be a partition of Ω into bounded polygonal sub-domains ν | Ω i smooth and ν ( x ) is s.p.d. for a.e. x ∈ Ω L. Agelas1, D. Di Pietro1, J. Droniou2, I. Kapyrin1, R. Eymard3, C. Guichard1, R. Masson1. Cell Centered Finite Volume Schemes for Multiphase Flow Applications

Outline Applications and meshes Flux formulation The L and G schemes Discrete variational framework Numerical Experiments Polyhedral admissible meshes T h : set of cells K E h = E i h ∪ E b h : set of inner and boundary faces σ m σ : surface of the face σ m K : volume of the cell K L. Agelas1, D. Di Pietro1, J. Droniou2, I. Kapyrin1, R. Eymard3, C. Guichard1, R. Masson1. Cell Centered Finite Volume Schemes for Multiphase Flow Applications

Outline Applications and meshes Flux formulation The L and G schemes Discrete variational framework Numerical Experiments Discrete function space V h V h : space of piecewise constant functions on T h v h ( x ) = v K for all x ∈ K Equip V h with the following discrete H 1 0 norm:   1 / 2  � � m σ | γ σ ( v h ) − v K | 2  ∀ v h ∈ V h , � v h � V h = d K ,σ K ∈T h σ ∈E K using the following trace reconstruction at the faces σ  γ σ ( v h ) = if σ = v K d L ,σ + v L d K ,σ  if σ = E K ∩ E L ∈ E i h ,  d L ,σ + d K ,σ   γ σ ( v h ) = 0 if σ ∈ E b h . L. Agelas1, D. Di Pietro1, J. Droniou2, I. Kapyrin1, R. Eymard3, C. Guichard1, R. Masson1. Cell Centered Finite Volume Schemes for Multiphase Flow Applications

Outline Applications and meshes Flux formulation The L and G schemes Discrete variational framework Numerical Experiments Finite Volume Scheme Let F K ,σ ( u h ) denote a conservative linear approximation of � ν ∇ u · n K ,σ σ σ = E K ∩ E L ∈ E i conservativity: F K ,σ ( u h ) + F L ,σ ( u h ) = 0 , h . The finite volume scheme reads find u h ∈ V h such that � � − F K ,σ ( u h ) = f for all K ∈ T h . K σ ∈E K L. Agelas1, D. Di Pietro1, J. Droniou2, I. Kapyrin1, R. Eymard3, C. Guichard1, R. Masson1. Cell Centered Finite Volume Schemes for Multiphase Flow Applications

Outline Applications and meshes Flux formulation The L and G schemes Discrete variational framework Numerical Experiments Discrete variational formulation For all u h , v h ∈ V h , let � � a h ( u h , v h ) = F K ,σ ( u h )( γ σ ( v h ) − v K ) K ∈T h σ ∈E K � � � = F K ,σ ( u h )( v L − v K ) − F K ,σ ( u h ) v K σ = E K ∩E L ∈E h K ∈T h σ ∈E K ∩E b h The finite volume scheme is equivalent to: find u h ∈ V h such that � a h ( u h , v h ) = fv h for all v h ∈ V h . Ω L. Agelas1, D. Di Pietro1, J. Droniou2, I. Kapyrin1, R. Eymard3, C. Guichard1, R. Masson1. Cell Centered Finite Volume Schemes for Multiphase Flow Applications

Outline Applications and meshes Flux formulation The L and G schemes Discrete variational framework Numerical Experiments Assumptions Flux consistency in Q : for all ϕ ∈ Q with ϕ h = { ϕ ( x K ) } K ∈T h , � � | F K ,σ ( ϕ h ) − m σ � ν ∇ ϕ � K · n K ,σ | 2 � � d K ,σ = 0 lim h → 0 m σ K ∈T h σ ∈E K Coercivity of the bilinear form a h : ∼ � v � 2 ∀ v ∈ V h , a h ( v , v ) > V h L. Agelas1, D. Di Pietro1, J. Droniou2, I. Kapyrin1, R. Eymard3, C. Guichard1, R. Masson1. Cell Centered Finite Volume Schemes for Multiphase Flow Applications