Finite Volume Schemes for multi-phase flow simulation on near well - PowerPoint PPT Presentation

Finite Volume Schemes for multi-phase flow simulation on near well grids J. Brac ( 1 ) , R. Eymard ( 2 ) , C. Guichard ( 1 , 2 ) and R. Masson ( 1 ) ( 1 ) IFP and ( 2 ) Université Paris-Est ECMOR XII - september 8th 2010 1

Objective Study of Finite Volume Schemes on 3D near well multi-phase flow simulations. Outline Applications and Difficulties � 3D Near Well Grids � Finite Volume Schemes � Numerical Experiments � Conclusion � 2

Multi-phase Flow Simulation on Near Well Grids Applications � Reservoir Simulation � CO2 geological storage Difficulties � Singular Pressure Distribution � Well-Radius << Reservoir Dimension � Deviated Well � Anisotropy 3

3D Near-well model Exponentially refined Unstructured mesh with Hybrid mesh with radial mesh only hexahedra hexahedra, tetrahedra and pyramids Discretization on complex 3D general meshes = ⇒ MultiPoint Flux Approximation (MPFA) Finite Volume Schemes 4

Finite Volume Scheme Model Problem Model Equation � −∇ · (Λ ∇ u ) = f in Ω , Find u ( potential ) in H 1 0 (Ω) | u = 0 on ∂ Ω bounded polygonal domain of R d Ω : Λ : symmetric positive definite tensor field function of L 2 (Ω) f : 5

Finite Volume Scheme Flux Formulation T h : set of cells K V h : space of piecewise constant functions on T h � • F K ,σ ( u h ) ≈ Λ ∇ u · n K ,σ linearly σ • Conservativity : F K ,σ ( u h ) + F L ,σ ( u h ) = 0 , σ = K | L Find u h ∈ V h | − � � σ ∈E K F K ,σ ( u h ) = K f ∀ K ∈ T h 6

Finite Volume Scheme Discrete Variational Formulation For all u h , v h ∈ V h , let � � � a h ( u h , v h ) = 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 | a h ( u h , v h ) = Ω fv h ∀ v h ∈ V h 7

Sample of MPFA Finite Volume Schemes O scheme [Aavatsmark et al., 1996, Edwards and Rogers, 1998] L scheme [Aavatsmark, 2007] G scheme inspired by the L scheme [Agélas et al., 2010a] → Subcell gradients satisfying continuity conditions → Subfluxes F G L ,σ → Convex linear combination F K ,σ = � θ G σ F G K ,σ → Choose the θ G σ to enhance the coercivity and remove singularities 8

The GradCell scheme uses a discrete variational formulation Non symmetric discrete variational formulation based on two cellwise constant gradients and residuals for stabilization � � � m σ m K Λ K ( ∇ h u h ) K · ( � a h ( u h , v h ) = ∇ h v h ) K + R K ,σ ( u h ) R K ,σ ( v h ) d K ,σ K ∈T h K ∈T h σ ∈E K � 1 ( ∇ h v h ) K = σ ∈E K m σ ( I K ,σ ( v h ) − v K ) n K ,σ m K � ( � 1 ∇ h v h ) K = σ ∈E K m σ ( γ σ ( v h ) − v K ) n K ,σ m K 9

The GradCell scheme has a compact stencil Fluxes are derived from the bilinear form. � � � a h ( u h , v h ) = 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 Fluxes F K ,σ ( u h ) only between cells sharing a face The stencil is compact: neighbours of the neighbours For topologically cartesian grids : 13 cells in 2D, 21 cells in 3D [Agélas et al., 2010b] 10

Outcome on Symmetry and Sparsity properties Fact � Previous schemes : compact but non symmetric ⇒ conditionnal coercivity � If GradCell symmetric ⇒ large stencil (81 cells in 3D) Difficult to combine both properties Wish � Symmetric unconditionally coercive scheme � Sparse stencil: 9 points in 2D and 27 points in 3D on topologically Cartesian meshes 11

SUSHI combining smart ideas... S cheme U sing S tabilization and H armonic I nterfaces combines... � O scheme ideas : subcell gradients ( ∇ h u ) s K and subfaces unknowns u s σ � A symmetric variational bilinear form for coercivity � Weak and consistent subcell gradient for convergence � Two point harmonic interpolation at the faces 12

SUSHI Nice harmonic interpolation formula Find a point y σ and a coefficient α with... � a linear two point interpolation... � ... exact on piecewise linear functions, � normal flux and potential continuity. Harmonic point y σ Harmonic interpolation u ( y σ ) = α u ( x K ) + ( 1 − α ) u ( x L ) [Agélas et al., 2009] 13

How SUSHI uses the interpolation formula ? At each face σ , choose the harmonic point y σ Subcell K s around a vertex s K s = ( x K , y σ , s , y σ ′ , x K ) 14

SUSHI Discrete subcell gradient 1 m Ks ( m s σ ( u s ( ∇ h u ) K s = σ − u K ) n K ,σ + m s σ ′ ( u s σ ′ − u K ) n K ,σ ′ + m ˆ σ ( u ˆ σ − u K ) n K s , ˆ σ + m ˆ σ ′ ( u ˆ σ ′ − u K ) n K s , ˆ σ ′ ) 15

SUSHI A symmetric formulation Symmetric discrete variational formulation � � � � � m K s m K s Λ K ( ∇ h u h ) s K · ( ∇ h v h ) s ( d K ,σ ) 2 R s K ,σ ( u h ) R s a h ( u h , v h ) = K + K ,σ ( v h ) K ∈T h s ∈V K σ ∈E K ∩E s Subfluxes � � � F s K ,σ ( u h )( v s a h ( u h , v h ) = σ − v K ) K ∈T h s ∈V K σ ∈E s ∩E K F s K ,σ ( u h ) + F s L ,σ ( u h ) = 0 16

Schemes comparison on single-phase analytical solution Problem studied � Single phase flow � Anisotropy of the tensor permeability Λ � Slanted well ⇒ Analytical solution [Aavatsmark and Klausen, 2003] Numerical study � O, L, G, GradCell and SUSHI schemes � Hexahedra and Hybrid mesh families � Λ = diag(1,1, 1 20 ) 17

Hexahedral mesh family l 2 pressure error mesh size h nonzero elements in the linear system � O and L schemes have the same behavior 18

Hybrid mesh family l 2 pressure error mesh size h nonzero elements in the linear system � GradCell stencil ≈ 4 times smaller than O scheme � L scheme fails but not the more flexible G scheme 19

Two-phase flow ( w - g ) near well simulation Injection of gaseous CO2 miscible in a reservoir full of water H2O ( w ) � Two-component CO2 ( w - g ) � Thermodynamic equilibrium defined by the solubility ¯ C � ( w − g ) CO2 < ¯ C w : C S g = 0 CO2 = ¯ C w ( g ) : C S g > 0 Test the O scheme on both types of meshes 20

Total Mass of CO2 function of time GOE = G rid O rientation E ffect 21

Mass of CO2 in phase gas function of time Cell size affect the oscillations 22

Conclusion � SUSHI scheme exhibits very promising results thanks to its unconditional coercivity � Hybrid meshes show drawbacks of the schemes : � O scheme has a stencil ≈ 4 times bigger than GradCell � L scheme fails but not the more flexible G scheme � Two-phase flow numerical solution is sensitive to GOE and size of the cells 23

References Aavatsmark, I. (2007). Multipoint flux approximation methods for quadrilateral grids. In 9th International Forum on Reservoir Simulation, Abu Dhabi . Aavatsmark, I., Barkve, T., Boe, O., and Mannseth, T. (1996). Discretization on non-orthogonal, quadrilateral grids for inhomogeneous, anisotropic media. J. Comput. Phys. , 127(1):2–14. Aavatsmark, I. and Klausen, R. (2003). Well index in reservoir simulation for slanted and slightly curved wells in 3d grids. SPE Journal , 8(1):41–48. Agélas, L., Di Pietro, D., and Droniou, J. (2010a). The g method for heterogeneous anisotropic diffusion on general meshes. M2AN Math. Model.Numer. Anal. Agélas, L., Di Pietro, D., Eymard, R., and Masson, R. (2010b). An abstract analysis framework for nonconforming approximations of anisotropic heterogeneous diffusion. International Journal on Finite Volumes , 7(1):1–29. Agélas, L., Eymard, R., and Herbin, R. (2009). A nine-point finite volume scheme for the simulation of diffusion in heterogeneous media. C. R. Math. Acad. Sci. Paris , 347(11-12):673–676. Edwards, M. and Rogers, C. (1998). Finite volume discretization with imposed flux continuity for the general tensor pressure equation. Computational Geosciences , 2(4):259–290. 24