Gradient Discretization of Hybrid Dimensional Darcy Flows in - PowerPoint PPT Presentation

Gradient Discretization of Hybrid Dimensional Darcy Flows in Fractured Porous Media Konstantin Brenner 1 , Maya Groza 1 , C. Guichard 2 , Gilles Lebeau 1 , Roland Masson 1 1 LJAD, Université Nice Sophia Antipolis, and team Coffee INRIA Sophia Antipolis 2 LJLL, Université Paris VI Séminaire de l’équipe EDP-MOISE-MGMI, LJK Grenoble 26 février 2015

Outline Darcy flows in Discrete Fractured Networks (DFN) Gradient scheme framework VAG and HFV schemes Two phase Darcy flows in DFN

Discrete Fracture Network (DFN) Full domain: Ω ⊂ R d , d = 2 , 3 Discrete Fracture Network: Γ = � i ∈ I Γ i Matrix domain: Ω \ Γ Σ = � i � = j ∂ Γ i ∩ ∂ Γ j Σ 0 = ∂ Γ ∩ ∂ Ω Σ N = ∂ Γ \ ∂ Ω

Hybrid dimensional models for DFN [Jaffré et al 2002] Integration in the fracture width d f << diam (Ω) d − 1 dimensional model in the fracture Continuous pressure assumption at the interface u + = u − = u f = γ u on Γ it assumes K f , n K m >> d f diam (Ω) Pressure continuity and flux conservation is assumed at fracture intersections

Hybrid dimensional models for Darcy flows in DFN: function spaces Pressure space V 0 : continuous at the matrix fracture and fracture fracture interfaces γ : H 1 (Ω) → L 2 (Γ) trace operator V 0 = { v ∈ H 1 0 (Ω) such that γ v ∈ H 1 (Γ) , γ v = 0 on ∂ Ω ∩ Γ } ,

Hybrid dimensional models for Darcy flows in DFN: function spaces Matrix and fracture flux space W :   q m ∈ H div (Ω \ Γ) ,    q f ∈ L 2 (Γ) d − 1 ,    W = , m · n + + q − m · n − ∈ L 2 (Γ) , such that div τ ( q f ) + q +     and normal flux conservation of q f at Σ \ Σ 0 in a weak sense   m · n + + q − Jump of the matrix normal flux on Γ : q + m · n − Main functional result: smooth function subspaces are dense in V 0 and W

Hybrid dimensional models for Darcy flows in DFN Find u ∈ V 0 , ( q m , q f ) ∈ W such that:  div ( q m ) = h m on Ω \ Γ ,      m · n + + q − m · n − = d f h f  div τ ( q f ) + q + on Γ ,    q m = − K m ∇ u on Ω \ Γ ,         q f = − d f K f ∇ τ γ u on Γ , 

Variational formulation in V 0 The weak formulation amounts to find u ∈ V 0 such that for all v ∈ V 0 one has: � � K m ∇ u · ∇ v d x + d f K f ∇ τ γ u · ∇ τ γ v d τ ( x ) Ω Γ � � = h m v d x + d f h f γ v d τ ( x ) . Ω Γ

Discretization: state of the art MFE or MHFE: Jaffré et al 2002, Firoozabadi 2008 TPFA: Karimi-Fard et al 2004 CVFE: Bastian et al 2006, Firoozabadi et al 2007 XFEM type methods: Formaggia, Scotti et al 2012 MPFA: Faille et al, Nordbotten et al, 2012, Edwards 2014 ... Our contributions: Extension of the Gradient scheme framework (Eymard et al 2010) to DFN models Convergence proof for single and two phase flow models Vertex Approximate Gradient (VAG) and Hybrid Finite Volume (HFV) discretizations

Gradient discretization framework: non conforming discretization Vector space of discrete unknowns: X 0 D Matrix and fracture gradient reconstruction operators D → L 2 (Ω) d ∇ D m : X 0 D → L 2 (Γ) d − 1 ∇ D f : X 0 Matrix and fracture function reconstruction operators: Π D m : X 0 D → L 2 (Ω) Π D f : X 0 D → L 2 (Γ) Assumption: � u D � D := �∇ D m u D � L 2 (Ω) d + �∇ D f u D � L 2 (Γ) d − 1 is a norm on X 0 D

Gradient discretization of the hybrid dimensional Darcy flow model u D ∈ X 0 D such that for all v D ∈ X 0 D one has � � K m ∇ D m u D · ∇ D m v D d x + d f K f ∇ D f u D · ∇ D f v D d τ ( x ) Ω Γ � � = h m Π D m v D d x + d f h f Π D f v D d τ ( x ) . Ω Γ Error estimate: �∇ D m u D − ∇ u � L 2 (Ω) d + �∇ D f u D − ∇ τ γ u � L 2 (Γ) d − 1 + � Π D m u D − u � L 2 (Ω) + � Π D f u D − γ u � L 2 (Γ) � � ≤ C ( C D , data ) S D ( u ) + W D ( q m , q f )

Gradient discretization of the hybrid dimensional Darcy flow model Coercivity : (discrete Poincaré inequality) � Π D m v D � L 2 (Ω) + � Π D f v D � L 2 (Γ) C D = max . � v D � D 0 � = v D ∈ X 0 D Consistency error : for all u ∈ V 0 � S D ( u ) = inf v D ∈ X 0 �∇ D m v D − ∇ u � L 2 (Ω) d + �∇ D f v D − ∇ τ γ u � L 2 (Γ) d − 1 D � + � Π D m v D − u � L 2 (Ω) + � Π D f v D − γ u � L 2 (Γ) , Conformity error : for all ( q m , q f ) ∈ W 1 �� W D ( q m , q f ) = sup ( ∇ D m v D · q m + (Π D m v D ) div ( q m ))( x ) d x � v D � D 0 � = v D ∈ X 0 Ω D � � m · n + + q − ( ∇ D f v D · q f + Π D f v D ( div τ ( q f ) + q + m · n − ))( x ) d τ ( x ) + , Γ

Vertex Approximate Gradient (VAG) and Hybrid Finite Volume (HFV) discretizations VAG HFV d.o.f. = cells v K , faces v σ , and d.o.f. = cells v K , fracture faces v σ , fracture edges v e and nodes v s ∇ D m v D : conforming P 1 FE ∇ D m v D : piecewise constant on a pyramidal submesh of K gradient on a tetrahedral submesh ∇ D f v D : conforming P 1 FE gradient ∇ D f v D : piecewise constant on triangular submesh of σ on a triangular submesh

VAG and HFV Fluxes VAG HFV Matrix fluxes: ∂ K = d.o.f. at the boundary of K T ν,ν ′ � F K ,ν ( v D ) = ( v K − v ν ′ ) K ν ∈ ∂ K Fracture fluxes: ∂σ = d.o.f. at the boundary of σ T ν,ν ′ � F σ,ν ( v D ) = ( v σ − v ν ′ ) σ ν ′ ∈ ∂ σ such that for all w D ∈ V 0 : � � K m ∇ D m v D · ∇ D m w D d x + d f K f ∇ D f v D · ∇ D f w D d τ Ω Γ � � � � = F K ,ν ( v D )( w K − w ν ) + F σ,ν ( v D )( w σ − w ν ) K ∈M ν ∈ ∂ K σ ∈F Γ ν ∈ ∂ σ

Choices of Π D m and Π D f : Control Volumes Π D m v D : piecewise constant on cell submeshes: � K = ω K ω K ,ν ν ∈ ∂ K \ dof Dir Π D f v D : piecewise constant on fracture face submeshes: � σ = ω σ ω σ,ν ν ∈ ∂σ \ dof Dir Practical choice is made to avoid the mixing of rocktypes at interfaces: Good choice: VAG1 Bad choice (CVFE like): VAG2 Nb: only the volume distribution coefficients are needed

Finite Volume Formulation of VAG and HFV: discrete conservation laws on control volumes � � Degrees of freedom: dof = M ∪ F Γ ∪ dof Dir ∪ dof \ ( M ∪ F Γ ∪ dof Dir ) � � F K ,ν ( u D ) = h m ( x ) d x , K ∈ M ω K ν ∈ ∂ K � � F σ,ν ( u D ) + − F K ,σ ( u D ) = ν ∈ ∂σ K ∈M σ � � � h f ( x ) d f ( x ) d τ ( x ) + h m ( x ) d x , σ ∈ F Γ , ω σ ω K ,σ K ∈M σ � � � � − F K ,ν ( u D ) + − F σ,ν ( u D ) = h m ( x ) d x ω K ,ν K ∈M ν σ ∈F Γ ,ν K ∈M ν � � + h f ( x ) d f ( x ) d τ ( x ) , ν ∈ dof \ ( M ∪ F Γ ∪ dof Dir ) , ω σ,ν σ ∈F Γ ,ν u ν = u ν , ν ∈ dof Dir .

VAG discretization : comparison with CVFE (Bastian et al 2006, Firozabadi et al 2007) Matrix fluxes local to the cell and fracture fluxes local to the face Coercive fluxes not depending on the choice of the control volumes Choice of the control volumes to respect material interfaces Still maintain a “nodal” approach due to the elimination of the cell unknowns VAG CVFE

Comparison of VAG and HFV schemes on hexahedral meshes Heterogeneous Isotropic Heterogeneous anisotropic

Comparison of VAG and HFV schemes on hexahedral meshes Isotropic case, Cartesian Anisotropic case, Cartesian Vertex Approximate Gradient Discretization Nb It F Err u Err g CR u CR g CPU It F Err u Err g CR u CR g CPU 1 · 10 − 4 8 · 10 − 4 1 3 1.2 0.04 0.11 n/a n/a 3 1.2 0.04 0.09 n/a n/a 7 · 10 − 3 6 · 10 − 3 2 5 2.1 0.01 0.03 1.89 1.62 5 1.9 0.01 0.02 2.12 1.87 2 · 10 − 3 1 · 10 − 2 2 · 10 − 3 6 · 10 − 3 8 · 10 − 2 3 9 2.4 1.92 1.71 0.01 9 2.2 2.06 1.83 7 · 10 − 4 3 · 10 − 3 5 · 10 − 4 2 · 10 − 3 4 16 2.5 1.95 1.77 0.9 14 2.1 2.03 1.81 0.7 2 · 10 − 4 8 · 10 − 4 1 · 10 − 4 5 · 10 − 4 5 30 2.5 1.97 1.82 9 20 2.2 2.02 1.84 5 5 · 10 − 3 2 · 10 − 4 3 · 10 − 5 1 · 10 − 4 6 56 2.5 1.89 1.86 90 29 2.2 2.01 1.87 48 Hybrid Finite Volume Discretization Nb It F Err u Err g CR u CR g CPU It F Err u Err g CR u CR g CPU 1.5 · 10 − 3 8 · 10 − 4 1 6 3.3 0.01 0.04 n/a n/a 4 2.5 0.01 0.16 n/a n/a 2 10 3.6 3 · 10 − 3 0.02 1.98 1.64 1.5 · 10 − 2 6 3.1 3 · 10 − 3 0.05 1.97 1.23 10 · 10 − 3 7 · 10 − 4 3 · 10 − 3 8 · 10 − 4 3 17 3.6 1.99 1.71 0.1 10 3.6 0.02 1.99 1.49 0.1 4 29 3.6 2 · 10 − 4 9 · 10 − 4 1.99 1.77 1.5 18 3.7 2 · 10 − 4 6 · 10 − 3 2.01 1.58 1.5 4 · 10 − 5 3 · 10 − 4 5 · 10 − 5 2 · 10 − 3 5 59 3.6 2 1.82 19 30 3.8 1.99 1.68 20 6 122 3.6 1 · 10 − 5 7 · 10 − 5 2 1.86 357 65 3.8 1 · 10 − 5 5 · 10 − 4 1.99 1.78 303

Comparison of VAG and HFV schemes on tetrahedral meshes Heterogeneous Isotropic Heterogeneous anisotropic