Finite Volume discretization of two phase Darcy flows with - PowerPoint PPT Presentation

Finite Volume discretization of two phase Darcy flows with discontinuous capillary pressures R. Eymard ( 1 ) , C. Guichard ( 2 ) , R. Herbin ( 3 ) , R. Masson ( 2 ) (1) Université Paris-Est (2) Université de Nice & INRIA Sophia Antipolis (3) Université d’Aix Marseille Siam Geosciences, june 17th-20th, 2013, Padova

Outline • Two phase Darcy flow in phase pressures formulation • Gradient scheme discretization • Convergence analysis • Vertex Approximate Gradient scheme • Numerical tests

Incompressible two-phase flow in heterogeneous porous media Equations for ( x , t ) in the space-time domain Ω × ( 0 , T ) M 1 ( x , S ( x , p c ( x , t )))Λ( x )( ∇ p 1 ( x , t ) − ρ 1 g ) = f 1 ( x , t )  � � Φ( x ) ∂ t S ( x , p c ( x , t )) − div      M 2 ( x , S ( x , p c ( x , t )))Λ( x )( ∇ p 2 ( x , t ) − ρ 2 g ) = f 2 ( x , t ) � � − Φ( x ) ∂ t S ( x , p c ( x , t )) − div     p c ( x , t ) = p 1 ( x , t ) − p 2 ( x , t )  Formulation in phase-pressures ( p 1 , p 2 ) with • p 1 pressure of the phase 1 (non wetting phase) • p 2 pressure of the phase 2 (wetting phase) • p c = p 1 − p 2 is the capillary pressure � • rocktypes: Ω = Ω j j ∈ J • S ( x , p c ) ∈ [ 0 , 1 ] is the saturation of the phase 1 with − S ( x , p c ) = S j ( p c ) for a.e. x ∈ Ω j and all p c ∈ R − S j is a non decreasing Lipschitz continuous function

Gradient scheme discretization [Eymard et al 2010] D = ( X D , Π D , ∇ D ) X D = R { d . o . f . } • discrete space ( X D , 0 with homogeneous Dirichlet BC) Π D : X D → L 2 (Ω) • reconstruction of function linear mapping ∇ D : X D → L 2 (Ω) d • reconstruction of gradient linear mapping such that � · � D = �∇ D · � L 2 (Ω) d is a norm on X D , 0 Examples of gradient schemes : − Conforming and Mixed Finite Elements − SUSHI and Mimetic schemes − Symmetric MPFA O scheme on tetrahedral meshes − VAG scheme

Discretization of the two-phase Darcy flow problem p 1 , ( n + 1 ) − ¯ p 2 , ( n + 1 ) − ¯ = p 1 , ( n + 1 ) − p 2 , ( n + 1 ) ∈ X D p ( n + 1 ) p 1 p 2 D ∈ X D , 0 D ∈ X D , 0 c s ( n + 1 ) ( x ) = S ( x , Π D p ( n + 1 ) ( x )) c D Φ( x ) s ( n + 1 ) ( x ) − s ( n ) � D ( x ) D Π D w ( x ) d x t ( n + 1 ) − t ( n ) Ω � M 1 ( x , s ( n + 1 ) ( x ))Λ( x )( ∇ D p 1 , ( n + 1 ) ( x ) − ρ 1 g ) · ∇ D w ( x ) d x + D Ω � t ( n + 1 ) 1 � f 1 ( x , t )Π D w ( x ) d x d t = ∀ w ∈ X D , 0 , ∀ n = 0 , . . . , N − 1 t ( n + 1 ) − t ( n ) t ( n ) Ω + equation for phase 2

Convergence analysis for two phase flow models • Global pressure formulation: • MFE-FE [Ewing et al 2001] • Finite Volume TPFA (Michel et al 2003) • Finite Volume Sushi, Mimetic [Brenner 2011], VAG [Brenner et al 2012], • Gradient schemes [Eymard et al 2013] • Phase by phase upwind scheme [Michel et al 2003]: only TPFA • Discontinuous capillary pressures [Cances et al 2012]: only TPFA Price to pay to extend the convergence analysis of the pressure pressure model to the Gradient scheme framework: ⋆ the approximation of the mobility is centered with M max ≥ M min > 0 M α ( x , s ) ∈ [ M min , M max ] ⋆ for ( x , s ) ∈ Ω × [ 0 , 1 ] ,

Properties of a sequence ( D m ) m ∈ N � Π D v � L 2 (Ω) • Coercivity C D = max ⇒ discrete Poincaré inequality � v � D v ∈ X D , 0 \{ 0 } C D m remains bounded • Consistency : S D m → 0 ∀ ϕ ∈ H 1 � � 0 (Ω) , S D ( ϕ ) = min � Π D v − ϕ � L 2 (Ω) + �∇ D v − ∇ ϕ � L 2 (Ω) d v ∈ X D , 0 • Limit-conformity : W D m → 0 � � 1 � � � ∀ ϕ ∈ H div (Ω) , W D ( ϕ ) = max ( ∇ D u ( x ) · ϕ ( x ) + Π D u ( x ) div ϕ ( x )) d x � � � u � D u ∈ X D , 0 \{ 0 } � � Ω � Π D v ( · + ξ ) − Π D v � L 2 ( R d ) ∀ ξ ∈ R d , T D ( ξ ) = • Compactness max � v � D v ∈ X D , 0 \{ 0 } | ξ |→ 0 sup lim T D m ( ξ ) = 0 m ∈ N

Convergence of the numerical scheme - sketch of proof • Estimates on �∇ D p α � L 2 (Ω × ( 0 , T )) d , α = 1 , 2 • Estimate on a dual semi-norm of the discrete time derivative of s D • Estimate on time and space translates of s D ⇒ strong convergence of s D in L 2 where p c = p 1 − p 2 • Minty trick : lim S ( ., Π D p c ) = S ( ., lim Π D p c ) • Convergence to the weak solution by consistency and limit-conformity

VAG scheme ( Vertex Approximate Gradient scheme ) X D = { discrete value u K at the cell centers x K and u s at the vertices s } • Tetrahedral submesh of each cell K 1 1 � � x s ′ x σ = Card V σ x s , u σ = Card V σ u s s ∈V σ s ∈V σ x K • Constant gradient on each tetrahedra T x σ � ( u s − u K ) g s ∇ T u = x s T s ∈V σ Piecewise constant gradient in L 2 (Ω) d ∇ D u = ∇ T u on each tetrahedra T Reconstruction operator Π D u ( x ) = u K on Ω K , u s on Ω Ks , with K = Ω K ∪ � s ∈V K \ ∂ Ω Ω K , s VAG is vertex-centered : unknowns u K can be eliminated from the linear system

Variational formulation and fluxes � � a D ( u , w ) = Λ( x ) ∇ D u ( x ) · ∇ D w ( x ) d x = f ( x ) Π D w ( x ) d x for all w ∈ X D , 0 . Ω Ω Finite Element nodal basis: η K , K ∈ M , η s , s ∈ V . V K : set of nodes of the cell K . �� �� � � � a D ( u , w ) = − Λ( x ) ∇ D u ( x ) · ∇ η s ( x ) d x w K − w s , K K ∈M s ∈V K � � � � = F K , s ( u ) w K − w s K ∈M s ∈V K with the fluxes � T s , s ′ � F K , s ( u ) = − Λ( x ) ∇ D u · ∇ η s ( x ) d x = ( u K − u s ′ ) . K K s ∈V K

Equivalent discrete conservation laws �  K 3 K 4 � F K , s ( u ) = f ( x ) d x for all K ∈ M ,     Ω K s ∈V K � � � − F K , s ( u ) = f ( x ) d x for all s ∈ V \ ∂ Ω  K 1  K 2   Ω K , s K ∈M s K ∈M s  � F K , s ( u ) = m K f ( x K ) for all K ∈ M ,    s ∈V K � � − F K , s ( u ) = m K , s f ( x K , s ) for all s ∈ V \ ∂ Ω    K ∈M s K ∈M s

Distribution of the volumes m K , s at the vertices The porous volume m K , s is taken from the surrounding cells proportionaly to the permeability of the cells K 3 K 4 s K 1 K 2

VAG discretization of the two phase flow model with upwinding F α K , s ( p α ) = F K , s ( p α ) + ρ α gF K , s ( z ) , α = 1 , 2 S and M α cellwise constant functions: S K , M α K , α = 1 , 2 S n K = S K ( p n S n K , s = S K ( p n c , K ) , c , s ) . κ 3 κ 4 S κ 3 , s S κ 3 , s  M α K ( S n K ) F α K , s ( p α, n ) if F α K , s ( p α, n ) ≥ 0 ,  G α K , s ( p 1 , n , p 2 , n ) = S κ 1 , s s M α K ( S n K , s ) F α K , s ( p α, n ) else  S κ 2 , s κ 1 κ 2 Equation for phase 1: K − S n − 1  S n � G α K , s ( p 1 , n , p 2 , n ) = 0 , K m K φ K + K ∈ M ,    ∆ t   s ∈V K S n K , s − S n − 1 � K , s � G α K , s ( p 1 , n , p 2 , n ) = 0 ,  m K , s φ K − s ∈ V \ ∂ Ω .   ∆ t   K ∈M s K ∈M s Similar equation for phase 2.

Problem of non uniqueness of the solution p 1 , p 2 Example: initial state with only phase 2: p c is not uniquely defined. To avoid this singularity when solving the discrete nonlinear system: Projections of p c , K on the interval: � � P c , K ( 0 ) , P c , K ( 1 ) and of p c , s on � � K ∈M s P c , K ( 0 ) , max min K ∈M s P c , K ( 1 ) .

Comparison with Control Volume Finite Element Methods Keep the cell center unknown Decouple the computation of fluxes from the choice of the Control volumes Fluxes always coercive Choice of the porous volumes to match heterogeneities Phase pressure unknowns to capture the jump of the saturations

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Oil migration in a 2D basin with 2 barriers

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Oil migration in a 2D anisotropic heterogeneous basin Density driven flow: ρ o = 850 , ρ w = 1000 kg / m 3 . � 0 . 82 � − 0 . 36 Permeability in the drain: Λ( x ) = 10 − 12 . − 0 . 36 0 . 28 Permeability in the barrier: Λ( x ) = 10 − 14 Id . Permeability in the fracture: Λ( x ) = 10 − 11 Id . k o r and k w r : Corey laws with S rw = 0 . 2, S ro = 0 and exponents 2. Capillary pressures for both rocktypes: Corey’s laws

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