Numerical Solution of Compositional Two-Phase Flow in Porous Media - PowerPoint PPT Presentation

Numerical Solution of Compositional Two-Phase Flow in Porous Media P. Bastian Collaborators: M. Blatt, O. Ippisch, R. Neumann Universit¨ at Heidelberg Interdisziplin¨ ares Zentrum f¨ ur Wissenschaftliches Rechnen Im Neuenheimer Feld 368, D-69120 Heidelberg email: Peter.Bastian@iwr.uni-heidelberg.de Linz, October 6, 2011 P. Bastian (IWR) Compositional Two-Phase Flow Linz, October 6, 2011 1 / 34

Overview Overview 1 Discontinuous Galerkin method for incompressible pure two-phase flow ◮ Implicit, fully-coupled ◮ Media discontinuities, internal Signorini-type interface conditions 2 Compositional two-phase flow with equilibrium phase exchange ◮ Phase disappearance problem, obstacle-type problem ◮ No switching, no additional variables 3 Software framework DUNE ◮ Rapid prototyping ◮ Many different meshes, adaptive refinement, dimension-independence ◮ Parallelization and efficient solvers P. Bastian (IWR) Compositional Two-Phase Flow Linz, October 6, 2011 2 / 34

DG for Two-Phase Flow DG for Two-Phase Flow Setting ◮ Both phases incompressible ◮ No dissolution, pure two-phase flow ◮ 1d, 2d, 3d, including gravity ◮ Different entry pressures in subdomains (media heterogeneities) Why DG ? ◮ Potentially higher order of convergence ◮ Full tensors ◮ Local conservation ◮ Handles elliptic, parabolic and hyperbolic equations ◮ nonconforming, unstructured grids / refinement P. Bastian (IWR) Compositional Two-Phase Flow Linz, October 6, 2011 3 / 34

DG for Two-Phase Flow Previous Work on DG for Two-Phase Flow Bastian & Rivi` ere ’2004 ◮ Global pressure / saturation formulation, splitting ◮ Implicit/explicit saturation(+limiters), H (div)-projection Eslinger ’2005 ◮ Splitting: Implicit Pressure, Implicit/explicit saturation ◮ Kirchhoff transformation, media disc., gravity, compressible Epshteyn & Rivi` ere ’2007 ◮ Fully implicit, Fully-coupled approach, p w , s n formulations ◮ no media discontinuities, no gravity, very coarse grids Ern, Mozolevski, Schuh ’2010 ◮ Splitting, global pressure, implicit saturation, H (div)-projection ◮ Media discontinuities, 1D, no gravity This work ◮ Fully- coupled, higher-order in time, p w , p c formulation, ◮ Media discontinuities, 1-3d P. Bastian (IWR) Compositional Two-Phase Flow Linz, October 6, 2011 4 / 34

DG for Two-Phase Flow p w , p c -Formulation P = { w , n } . Two coupled equations for p w , p c : −∇ · { ( λ w + λ n ) K ∇ p w + λ n K ∇ p c − ( λ w ρ w + λ n ρ n ) Kg } = f ∂ t { φ (1 − ψ ( p c )) } − ∇ · { λ n K ( ∇ p w + ∇ p c − ρ n g ) } = f n Nonlinearities: λ w ( p c ) = k rw ( ψ ( p c )) λ n ( p c ) = k rn (1 − ψ ( p c )) s w = ψ ( p c ) , , . µ w µ n s w 1 1 k rn k rw ψ ( p c ) 0 0 p e p c s w 0 1 p c < p e : s n = 0 , ∂ t ( φ s n ) = 0 , λ n = 0 → singular! Allow ψ ( p c ) > 1. P. Bastian (IWR) Compositional Two-Phase Flow Linz, October 6, 2011 5 / 34

DG for Two-Phase Flow Media Discontinuities Two regions with different ψ ( p c ) curves: s w ψ ( l ) ψ ( h ) 1 s ∗ ψ ( l ) ψ ( h ) w n F 0 p c p ( l ) p ( h ) Ω e e Interface conditions ( van Duijn et al., ’1995 ): � p ( l ) p ( l ) ≥ p ( h ) c c e p ( h ) = c p ( h ) else e At the media interface let n F point into the ( l ) region and set � p ( h ) − p ( l ) p ( l ) ≥ p ( h ) c c c e J ( p c ) = p ( h ) − p ( h ) else c e P. Bastian (IWR) Compositional Two-Phase Flow Linz, October 6, 2011 6 / 34

DG for Two-Phase Flow DG Formulation: Pressure Equation Setting u = − ( λ w + λ n ) K ∇ p w − λ n K ∇ p c + ( λ w ρ w + λ n ρ n ) Kg , then � � � � − u · ∇ v + n F · { u } ω � v � + ( γ F � v � − θ n F · { λ K ∇ v } ω ) � p w � T ∈T h F ∈F i , D T F h � � � � � � ( γ F v − θ n F · ( λ K ∇ v ))) p D = fv − jv + w T ∈T h F ∈F N F ∈F D T F h h Weighted averages ( Di Pietro, Ern, Guermond ’2008 ): δ + δ − ω − = ω + = δ ± Kn = n t F K ± n F , Kn Kn , . δ − Kn + δ + δ − Kn + δ + Kn Kn Penalty term ( �·� denotes harmonic average): | F | γ F = � λδ Kn � M , M = α k ( k + n − 1) min( | T − ( F ) | , | T + ( F ) | ) P. Bastian (IWR) Compositional Two-Phase Flow Linz, October 6, 2011 7 / 34

DG for Two-Phase Flow DG Formulation: Saturation Equation Setting u n = λ n q n , q n = − K ( ∇ p w + ∇ p c − ρ n g ), then   � � �  ∂ t φ (1 − ψ ( p c )) w − u n · ∇ w  T ∈T h T T � � λ ↑ n n F · { q n } ω � w � + ( γ n , F � w � − θλ ↑ + n n F · { K ∇ w } ω ) J ( p c ) F ∈F i , Dn F h � � � � � � ( γ n , F w − θλ n n F · ( K ∇ w ))) p D = f n w − j n w + c T ∈T h F ∈F Nn F ∈F Dn T F h h Upwinding of mobility: 2 k − rn ( s − p − n ) k + rn ( s + � n ) n F · { q n } ω ≥ 0 p ↑ s ± n = 1 − ψ ± ( p ↑ λ ↑ c c = , c ) , n = . p + else ( k − rn ( s − n ) + k + rn ( s + n )) µ n c Penalty parameter: γ n , F = λ − n + λ + γ n , F = λ ↑ n n � δ Kn � M , F ∈ Γ , � δ Kn � M , else . 2 P. Bastian (IWR) Compositional Two-Phase Flow Linz, October 6, 2011 8 / 34

DG for Two-Phase Flow A Model Problem P. Bastian (IWR) Compositional Two-Phase Flow Linz, October 6, 2011 9 / 34

DG for Two-Phase Flow Grid Convergence Study Parameters ◮ Brooks Corey capillary pressure ◮ Quadratic relative permeability ◮ scalar absolute permeability ◮ P 1 on quadrilaterals, 2nd order Alexander scheme in time ◮ P 2 on quadrilaterals, 3rd order Alexander scheme in time Reference scheme ◮ Cell-centered finite volume scheme, two-point flux approximation, implicit Euler ◮ p w , p c formulation ◮ Upwinding of mobility, as described above Meshes and timesteps ◮ From 40 × 24, ∆ t = 60 s to 1280 x 768, ∆ t = 1 . 875, T = 1800 s , 3600 s ◮ 2 million DOF, 2000 time steps for CCFV reference solution P. Bastian (IWR) Compositional Two-Phase Flow Linz, October 6, 2011 10 / 34

DG for Two-Phase Flow T=1800s P. Bastian (IWR) Compositional Two-Phase Flow Linz, October 6, 2011 11 / 34

DG for Two-Phase Flow T=1800s, Zoom P. Bastian (IWR) Compositional Two-Phase Flow Linz, October 6, 2011 12 / 34

DG for Two-Phase Flow T=3600s, Zoom1 P. Bastian (IWR) Compositional Two-Phase Flow Linz, October 6, 2011 13 / 34

DG for Two-Phase Flow T=3600s, Zoom2 P. Bastian (IWR) Compositional Two-Phase Flow Linz, October 6, 2011 14 / 34

DG for Two-Phase Flow Same Problem in 3d P. Bastian (IWR) Compositional Two-Phase Flow Linz, October 6, 2011 15 / 34

DG for Two-Phase Flow Conclusion DG scheme Number of time steps (Newton convergence) in 2d: T = 1800s T = 3600s h − 1 ∆ t CCFV DG CCFV DG 40 60.000 30 30 60 60 80 30.000 60 60 120 120 160 15.000 120 120 240 240 320 7.500 240 260 557 566 640 3.750 480 960 1280 1.875 960 1925 Conclusion: Fully coupled DG scheme runs robustly and agrees with CCFV scheme Saves about two levels of refinement in space and time P. Bastian (IWR) Compositional Two-Phase Flow Linz, October 6, 2011 16 / 34

Compositional Two-Phase Flow Compositional Two-Phase Flow Consider two phases P = { l , g } , two components C = { w , g } and equilibrium phase exchange. Mole conservation for each component: φ∂ t ( ν l x w l s l + ν g x w g s g ) + ∇ · ( ν l x w l u l + ν g x w g u g + j w l + j w g ) = f w φ∂ t ( ν l x g g s g ) + ∇ · ( ν l x g g u g + j g l s l + ν g x g l u l + ν g x g l + j g g ) = f g Extended Darcy’s law and diffusion terms: j κ α = − ( τφ s α D κ α ) ν α ∇ x κ j w α + j g u α = − λ α K ( ∇ p α − ρ α g ) , α , α = 0 . Capillary pressure: p c = p g − p l , s l = ψ ( p c ) , s l + s g = 1 . Solubility (given in parametrized form): x g x w l = 1 − x g x w x g g = 1 − x w l = ξ ( p g , T , . . . ) , l , g = η ( p g , T , . . . ) , g . P. Bastian (IWR) Compositional Two-Phase Flow Linz, October 6, 2011 17 / 34

Compositional Two-Phase Flow Phase Disappearance Problem If both phases are present, 0 < s l , s g < 1, two variables describe the state of the system, e.g ( p β , s n ) or ( p β , p c ). If s g = 0 (gas phase disappears) then the balance equations reduce to l K ( ∇ p l − ρ l g )) = f w + f g φ∂ t ν l − ∇ · ( ν l µ − 1 φ∂ t ( ν l x g l ) + ∇ · ( ν l x g l u l + j g l ) = f g → Single phase flow + transport problem. Variables ( p l , x g l ) describe the state of the system. Note that x g l is now an independent variable. Several ways to overcome this: Variable switching ( Class, ’2000 ) Complementarity conditions ( Jaffr´ e, Sboui, ’2010 ) Extended formulations ( Abadbour & Panfilov, ’2009 , Bourgeat et al., preprint ) P. Bastian (IWR) Compositional Two-Phase Flow Linz, October 6, 2011 18 / 34