Stochastic multiscale modeling of subsurface and surface flows. Part - PowerPoint PPT Presentation

Stochastic multiscale modeling of subsurface and surface flows. Part I: Multiscale mortar mixed finite elements for Darcy flow Ivan Yotov Department of Mathematics, University of Pittsburgh KAUST WEP Workshop January 30–February 1, 2010 Joint work with Todd Arbogast , Gergina Pencheva , Sunil Thomas , and Mary F. Wheeler , The University of Texas at Austin; Benjamin Ganis , University of Pittsburgh Department of Mathematics, University of Pittsburgh 1

Energy and environment • Ground water and surface water contamination • Hydrocarbon energy production Department of Mathematics, University of Pittsburgh 2

Reservoir rock Department of Mathematics, University of Pittsburgh 3

Mathematical and numerical challenges for modeling porous media • Multiscale - space and time • Multiphysics - aquifer, surface water, waterflood, CO 2 , polymer, geomechanics • Multiphase - gas, aqueous, multiple flowing phases • Highly nonlinear coupled PDE systems - advection, reaction, diffusion/dispersion, capillary effects • Complex geology and geometry - faults, fractures, layers • Very large scale computations – millions of unknowns – parallel computing Department of Mathematics, University of Pittsburgh 4

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Groundwater flow in a faulted aquifer Numerical grids and low conductivity barriers Z Y X Department of Mathematics, University of Pittsburgh 6

Contaminated groundwater flow DNAPL concentration at 44 days Z Y X Department of Mathematics, University of Pittsburgh 7

Outline • Background and motivation • A multiscale mortar mixed finite element method • A priori error estimates • A domain decomposition algorithm • A posteriori error estimates • Mortar and subdomain adaptivity • A relationship between multiscale mortar MFE methods and subgrid upscaling methods • A multiscale flux basis formulation • Extension to two-phase flow Department of Mathematics, University of Pittsburgh 8

Single phase flow model in Ω ⊂ R d ( d = 2 , 3) u = − K ∇ p (Darcy’s law) ∇ · u = f in Ω (conservation of mass) u · n = 0 on ∂ Ω (no flow BC) Variational mixed formulation H ( div ; Ω) = { v : v ∈ ( L 2 (Ω)) d , ∇ · v ∈ L 2 (Ω) } V = { v ∈ H ( div ; Ω) : v · n = 0 on ∂ Ω } � W = L 2 0 (Ω) = { w ∈ L 2 (Ω) : w dx = 0 } . Ω Find u ∈ V , p ∈ W such that ( K − 1 u , v ) = ( p, ∇ · v ) , v ∈ V , ( ∇ · u , w ) = ( f, w ) , w ∈ W. Department of Mathematics, University of Pittsburgh 9

✂✄ ✁ ✄ ✂ � �✁ The mixed finite element method T h - finite element partition pressure velocity V h × W h ⊂ V × W - mixed finite element spaces Find u h ∈ V h , p h ∈ W h such that ( K − 1 u h , v ) = ( p h , ∇ · v ) , v ∈ V h , ( ∇ · u h , w ) = ( f, w ) , w ∈ W h . • Simultaneous (accurate) approximation of pressure and velocity • Local mass conservation: for each element E, � � � 1 on E, w = = ⇒ ∇ · u h = q. 0 otherwise E E • Continuity of normal flux across element faces: for each e = ∂E 1 ∩ ∂E 2 , u h | E 1 · n e = u h | E 2 · n e . Department of Mathematics, University of Pittsburgh 10

Error estimates and convergence testing Theorem [Ingram-Wheeler-Xue-Y.]: � u − u h � + �∇ · ( u − u h ) � + � p − p h � ≤ Ch pres errp 2.8 0.007 2.6 0.0065 2.4 0.006 2.2 0.0055 2 0.005 1.8 0.0045 1.6 0.004 1.4 0.0035 1.2 0.003 1 0.0025 0.8 0.002 0.6 0.0015 0.4 0.001 0.2 0.0005 Computed pressure and velocity Pressure and velocity error R h R h R h 1 /h � p − p h � � u − u h � �∇ · ( u − u h ) � p u ∇· u 8 0.120E0 0.164E0 0.188E0 16 0.605E-1 1.0 0.834E-1 1.0 0.941E-1 1.0 32 0.304E-1 1.0 0.417E-1 1.0 0.470E-1 1.0 64 0.152E-1 1.0 0.208E-1 1.0 0.235E-1 1.0 Department of Mathematics, University of Pittsburgh 11

Motivation for multiscale modeling: flow in heterogeneous porous media Heterogeneous permeability varies on a fine scale. Full fine scale grid resolution ⇒ large, highly coupled system of equations ⇒ solution is computationally intractable • Variational Multiscale Method – Hughes et al; Brezzi – Mixed FEM: Arbogast et al • Multiscale Finite Elements – Hou, Wu, Cai, Efendiev et al – Mixed FEM: Chen and Hou; Aarnes et al New approach: based on domain decomposition and mortar finite elements More flexible - easy to improve global accuracy by refining the local mortar grid where needed Department of Mathematics, University of Pittsburgh 12

Multiscale finite element/subgrid upscaling methods L ǫ u = f ⇒ u ∈ V : a ( u, v ) = ( f, v ) ∀ v ∈ V. Multiscale approximation: H - coarse grid, h ≈ ǫ - fine grid (subgrid) h � � � V H,h = V H + V ′ h � � � � � h ( E ) : φ E Basis for V ′ h,i , i = 1 , . . . , N E , � � � � � H a E ( φ E H,i + φ E h,i , v h ) = 0 ∀ v h ∈ V h ( E ) � � � � � � � � Multiscale solution: u H,h ∈ V H,h , a ( u H,h , v H,h ) = ( f, v H,h ) ∀ v H,h ∈ V H,h Department of Mathematics, University of Pittsburgh 13

Multiblock formulation for single phase flow ¯ i =1 ¯ Ω = ∪ n Ω i ; Γ ij = ∂ Ω i ∩ ∂ Ω j On each block Ω i : u = − K ∇ p in Ω i ∇ · u = q in Ω i u · n = 0 on ∂ Ω i ∩ ∂ Ω On each interface Γ ij : p i = p j on Γ ij [ u · n ] ij = 0 on Γ ij where p i = p | ∂ Ω i [ u · n ] ij ≡ u | Ω i · n − u | Ω j · n Department of Mathematics, University of Pittsburgh 14