Multipoint Flux Mixed Finite Element Method in Porous Media - PowerPoint PPT Presentation

Multipoint Flux Mixed Finite Element Method in Porous Media Applications Part I: Introduction and Multiscale Mortar Extension Guangri Xue (Gary) KAUST GRP Research Fellow Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin In collaboration with: Mary F. Wheeler, The University of Texas at Austin Ivan Yotov, University of Pittsburgh Acknowledgement: GRP Research Fellowship, made by KAUST KAUST WEP Workshop, Saudi Arabia, 1/30/2010 Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA

Modeling Carbon Sequestration Key Processes • CO 2 /brine mass transfer • Multiphase flow • During injection (pressure driven) • After injection (gravity driven) • Geochemical reactions • Geomechanical modeling Numerical Simulations • Characterization (fault, fractures) • Appropriate gridding • Compositional EOS • Parallel computing capability Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA

Corner Point Geometry • General hexahedral grid (with non-planar faces) • Fractures and faults • Pinch-out • Layers • Non-matching Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA

Outline • Some locally conservative H(div) conforming method • Multipoint flux mixed finite element method (MFMFE) • Multiscale Mortar MFMFE • Numerical examples • Summary and Conclusions Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA

Some locally conservative H(div) conforming method • Mixed Finite Element Raviart, Thomas 1977; Nedelec 1980; Brezzi, Douglas, Marini 1985; Brezzi, Douglas, Duran, Fortin 1987; Brezzi, Douglas, Duran, Marini 1985; Chen, Douglas 1989, Shen 1994; Kuznetsov, Repin 2003; Arnold, Boffi, Falk 2005; Sbout, Jaffre, Roberts 2009... • Mimetic Finite Difference Shashkov, Berndt, Hall, Hyman, Lipnikov, Morel, Moulton, Roberts, Steinberg, Wheeler, Yotov ... • Cell-Centered Finite Difference Russell, Wheeler 1983; Arbogast, Wheeler, Yotov 1997; Arbogast, Dawson, Keenan, Wheeler, Yotov 1998 ... • Multipoint Flux Approximation Aavatsmark, Barkve, Mannseth 1998; Aavatsmark 2002; Edwards 2002; Edwards, Rogers 1998, ... • Multipoint Flux MFE Wheeler, Yotov 2006; Ingram, Wheeler, Yotov 2009; Wheeler, X., Yotov 2009, 2010 Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA

Multipoint Flux Mixed Finite Element (MFMFE)—1 Find u ∈ H (div), p ∈ L 2 , ( K − 1 u , v ) − ( p, ∇ · v ) = 0 , ∀ v ∈ H (div) ∀ q ∈ L 2 ( ∇ · u , q ) = ( f, q ) , MFMFE method: find u h ∈ V h , p h ∈ W h , ( K − 1 u h , v ) Q − ( p, ∇ · v ) = 0 , ∀ v ∈ V h ( ∇ · u , q ) = ( f, q ) , ∀ q ∈ W h Finite element space: V h ( E ) and W h ( E ) � � � � v ∈ ˆ V ( ˆ q ∈ ˆ W ( ˆ V h ( E ) = P ˆ v | ˆ E ) , W h ( E ) = q | ˆ E ) Numerical quadrature rule: � 1 � ( K − 1 u h , v h ) Q = ( K − 1 u h , v h ) Q,E = J B T K − 1 B ˆ � � u h , ˆ v h Q, ˆ E E ∈T h E ∈T h Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA

Multipoint Flux Mixed Finite Element (MFMFE)—2 FEM space on ˆ E : • Simplicial element [Brezzi, Douglas, Marini 1985; Brezzi, Douglas, Duran, Fortin 1987] : E ) d , ˆ V ( ˆ E ) = P 1 ( ˆ W ( ˆ ˆ E ) = P 0 ( ˆ E ) , • 2D square [Brezzi, Douglas, Marini 1985] : α 1 x + β 1 y + r 1 + rx 2 + 2 sxy � � V ( ˆ ˆ E ) = BDM 1 ( ˆ E ) = α 2 x + β 2 y + r 2 − 2 rxy − sy 2 W ( ˆ ˆ E ) = P 0 ( ˆ E ) • 3D cube [Ingram, Wheeler, Yotov 2009] : z ) T + r 3 curl(0 , 0 , ˆ x 2 ˆ x 2 ˆ z ) T V ( ˆ ˆ E ) = BDDF 1 ( ˆ E ) + r 2 curl(0 , 0 , ˆ y ˆ y 2 , 0 , 0) T + s 3 curl(ˆ y 2 ˆ z, 0 , 0) T + s 2 curl(ˆ x ˆ x ˆ z 2 , 0) T + t 3 curl(0 , ˆ z 2 , 0) T + t 2 curl(0 , ˆ y ˆ x ˆ y ˆ W ( ˆ ˆ E ) = P 0 ( ˆ E ) Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA

Multipoint Flux Mixed Finite Element (MFMFE)—3 Numerical quadrature rule on ˆ V ( ˆ E ) : � 1 � ( K − 1 u h , v h ) Q,E = J B T K − 1 B ˆ u h , ˆ v h Q, ˆ E Symmetric [Wheeler and Yotov 2006] : n v = | ˆ � 1 � 1 E | � � J B T K − 1 B ˆ J B T K − 1 B ˆ � u h , ˆ u h · ˆ | ˆ v h v h r i n v Q, ˆ E i =1 n v : number of vertices of ˆ E . Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA

Properties of MFMFE—1 � N T � � � � � U 0 M = 0 P − F N Basis functions in ˆ V ( ˆ E ): v 11 ( r 1 ) · n 1 = 1 , v 11 ( r 1 ) · n 2 = 0 v 11 ( r i ) · n j = 0 , for i � = 1 , j = 1 , 2 � 1 � J B T K − 1 Bv 11 , v 11 � = 0 Q, ˆ E � 1 � J B T K − 1 Bv 11 , v 12 � = 0 Q, ˆ E � 1 � J B T K − 1 Bv 11 , v ij = 0 , i � = 1 Q, ˆ E M is block diagonal. Cell-centered scheme: NM − 1 N T P = F Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA

Properties of MFMFE—2 • Locally conservative • Cell-centered scheme, ”solver friendly” • Equivalent to multipoint flux approximation method • Accurate for full tensor coefficient, simplicial grids, h 2 -quadrilateral grid, and h 2 -hexahedral grid with non-planar faces • Superconvergent Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA

Convergence Results of MFMFE Symmetric method Theorem [Wheeler and Yotov 2006, Ingram, Wheeler, and Yotov 2009] On simplicial grids, h 2 -parallelograms, and h 2 -parallelepipeds � u − u h � + � div( u − u h ) � + � p − p h � ≤ Ch � Q h p − p h � ≤ Ch 2 , for regular h 2 -parallelpipeds Proposition On h 2 -parallelogram and K -orthogonal grids, � Π R u − Π R u h � ≤ Ch 2 Π R : RT 0 projection Open question for non-orthogonal grid. Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA

Multiscale Mortar MFMFE Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA

Multidomain variational formulation n � V i = H (div; Ω i ) , V = V i , i =1 n W i = L 2 (Ω i ) , W i = L 2 (Ω) . � W = i =1 Λ i,j = H 1 / 2 (Γ i,j ) , � Λ = Λ i,j . 1 ≤ i<j ≤ n Find u ∈ V , p ∈ W , and λ ∈ Λ such that, for 1 ≤ i ≤ n , ( K − 1 u , v ) Ω i − ( p, ∇ · v ) Ω i = −� g, v · n i � ∂ Ω i / Γ − � λ, v · n i � Γ i , ∀ v ∈ V i , ( ∇ · u , w ) Ω i = ( f, w ) Ω i , ∀ w ∈ W i , n � � u · n i , µ � Γ i = 0 , ∀ µ ∈ Λ . i =1 Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA

Multiscale Mortar MFMFE: formulation n n Γ ij spaces � � V h = W h = V h,i , W h,i Ω i Ω j i =1 i =1 olynomial � Λ H = Λ H,i,j olynomial 1 ≤ i<j ≤ n Multiscale mortar MFMFE method is defined as: seek u h ∈ V h , p h ∈ W h , λ H ∈ Λ H such that for 1 ≤ i ≤ n , ( K − 1 u h , v ) Q, Ω i − ( p h , ∇ · v ) Ω i = − � g, Π R v · n i � ∂ Ω i / Γ − � λ H , Π R v · n i � Γ i , ∀ v ∈ V h,i , ( ∇ · u h , w ) Ω i = ( f, w ) Ω i , ∀ w ∈ W h,i , n � � Π R u h · n i , µ � Γ i = 0 , ∀ µ ∈ Λ H . ) i =1 Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA

Multiscale Mortar MFMFE: an interface formulation—1 Interface problem: d H ( λ H , µ ) = g H ( µ ) , µ ∈ Λ H , d H : L 2 (Γ) × L 2 (Γ) → R for λ, µ ∈ L 2 (Γ) by n n � Π R u ∗ � � d H ( λ, µ ) = d H,i ( λ, µ ) = − h ( λ ) · n i , µ � Γ i . i =1 i =1 g H : L 2 (Γ) → R : n n � � g H ( µ ) = g H,i ( µ ) = � Π R ¯ u h · n i , µ � Γ i , i =1 i =1 Star problem: ( u ∗ h ( λ ) , p ∗ h ( λ )) ∈ V h × W h solve, for 1 ≤ i ≤ n , ( K − 1 u ∗ h ( λ ) , v ) Q, Ω i − ( p ∗ h ( λ ) , ∇ · v ) Ω i = −� λ, Π R v · n i � Γ i , v ∈ V h,i , ( ∇ · u ∗ h ( λ ) , w ) Ω i = 0 , w ∈ W h,i . Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA

Multiscale Mortar MFMFE: an interface formulation—2 Bar problem: (¯ u h , ¯ p h ) ∈ V h × W h solve, for 1 ≤ i ≤ n , ( K − 1 ¯ u h ( λ ) , v ) Q, Ω i − (¯ p h ( λ ) , ∇ · v ) Ω i = −� g, Π R v · n i � ∂ Ω i / Γ i , v ∈ V h,i , ( ∇ · ¯ u h ( λ ) , w ) Ω i = 0 , w ∈ W h,i . with u h = u ∗ p h = p ∗ h ( λ H ) + ¯ h ( λ H ) + ¯ u h , p h . Center for Subsurface Modeling Institute for Computational Engineering and Sciences The University of Texas at Austin, USA