Numerical Upscaling and Preconditioning of Flows in Highly - PowerPoint PPT Presentation

Introduction Discretization Preconditioning of the Fine Grid Systems Numerical Upscaling and Preconditioning of Flows in Highly Heterogeneous Porous Media R. Lazarov, TAMU, Y. Efendiev, J. Galvis, and J. Willems WS#4: Numerical Analysis of Multiscale Problems & Stochastic Modelling RICAM, Linz, Dec. 12-16, 2011 Thanks: NSF, KAUST 1 / 51

Introduction Discretization Preconditioning of the Fine Grid Systems Outline Introduction 1 Motivation and Problem Formulation Discretization 2 Single Grid Approximation Subgrid Approximation and Its Performance Preconditioning of the Fine Grid Systems 3 Overlapping DD method Some Numerical Examples When one can use this method ? Examples 2 / 51

Introduction Discretization Motivation and Problem Formulation Preconditioning of the Fine Grid Systems Outline Introduction 1 Motivation and Problem Formulation Discretization 2 Single Grid Approximation Subgrid Approximation and Its Performance Preconditioning of the Fine Grid Systems 3 Overlapping DD method Some Numerical Examples When one can use this method ? Examples 3 / 51

Introduction Discretization Motivation and Problem Formulation Preconditioning of the Fine Grid Systems Motivation: media at multiple scales Figure: Porous media: real-life scale and macro scale 4 / 51

Introduction Discretization Motivation and Problem Formulation Preconditioning of the Fine Grid Systems Motivation: industrial foams – media of low solid fraction Figure: Industrial foams on micro-scale; porosity over 93% Figure: Trabecular bone: micro- and macro-scales 5 / 51

Introduction Discretization Motivation and Problem Formulation Preconditioning of the Fine Grid Systems Modeling of flow in porous media (1) Flows in porous media are modeled by linear Darcy law that relates the macroscopic pressure p and velocity u : ∇ p = − µκ − 1 u , κ − permeability , µ − viscosity (1) (2) Another venue for a two-phase flow is a Richards model: ∇ p = − µκ − 1 u , where κ = k ( x ) λ ( x , p ) (2) with k ( x ) heterogeneous function is the intrinsic permeability, while λ ( x , p ) is a smooth function that varies moderately in both x and p , related to the relative permeability. (3) For flows in highly porous media Brinkman (1947) enhanced Darcy’s law by adding dissipative term scaled by viscosity: ∇ p = − µκ − 1 u + µ ∆ u . (3) 6 / 51

Introduction Discretization Motivation and Problem Formulation Preconditioning of the Fine Grid Systems Computer Generated Heterogeneities Distributions Allaire in 1991 studied homogenization of slow viscous fluid flows (with negligible no-slip effects on interface between the fluid and the solid obstacles) for periodic arrangements. Computer generated permeabilty fields are shown below. Figure: periodic + rand; rand + rand, SPE10 slice, fractured 7 / 51

Introduction Discretization Motivation and Problem Formulation Preconditioning of the Fine Grid Systems The Goals of Our Research Our aim is development and study of numerical method that address the following main issues of the above classes of problem: Works well in both limits, Darcy and Brinkman; Applicable to linear and nonlinear highly heterogeneous problems; Could be used as a stand alone numerical upscaling procedure; Is robust with respect to high variations of the permeability field. 8 / 51

Introduction Discretization Motivation and Problem Formulation Preconditioning of the Fine Grid Systems State of the Art: Available Numerical Methods and Tools Numerical upscaling as subgrid approximation methods: (1) standard FEM for Darcy: Hou & Wu, 1997, Wu, Efendiev, & Hou, 2002, comprehensive exposition in the book of Efendiev & Hou, 2009, (2) mixed methods for Darcy: Aarnes, 2004, Arbogast, 2002, Chen & Hou, 2002, (3) DG FEM for Brinkman equation: Willems, 2009, Iliev, Lazarov & Willems, 2010, Juntunen & Stenberg, 2010, Könö & Stenberg, 2011 (4) Galerkin FEM augmented with multiscale finite element functions: Pechstein & Scheichl, 2008, 2011, Chu, Graham, & Hou, 2010. 9 / 51

Introduction Discretization Motivation and Problem Formulation Preconditioning of the Fine Grid Systems State of the Art: Available Numerical Methods and Tools Preconditioning techniques based on coarse grid space with multiscale finite element functions: (1) FETI and other DD methods: Graham, Klie, Lechner, Pechstein, & Scheichl, 2007, 2008, 2009, 2011 (2) Using energy-minimizing coarse spaces: Xu & Zikatanov, 2004 (3) Spectral Element Agglomerate Algebraic Multigrid Methods, Efendiev, Galvis, & Vassielvski, 2011 (4) Use of multiscale basis in DD, Aarnes and Hou (2002) 10 / 51

Introduction Discretization Motivation and Problem Formulation Preconditioning of the Fine Grid Systems Objectives: Derive, study, implement, and test a numerical upscaling procedure for 1 highly porous media that works well in both limits, Brinkman and Darcy, so it could cover both, natural porous media and man-made materials; Design and study of preconditioning techniques for such problems for 2 porous media of high contrast with targeted applications to oil/water reservoirs, bones, filters, insulators, etc 11 / 51

Introduction Discretization Motivation and Problem Formulation Preconditioning of the Fine Grid Systems Strategy: Multiscale finite element method 1 Mixed and Galerkin formulations 2 Coarse-grid finite element spaces augmented with fine-scale functions 3 based on local weighted spectral problems (after Efendiev Galvis) Robust with respect to the contrast iterative techniques for solving large 4 fine grid systems; Experimentation 5 12 / 51

Introduction Single Grid Approximation Discretization Subgrid Approximation and Its Performance Preconditioning of the Fine Grid Systems Outline Introduction 1 Motivation and Problem Formulation Discretization 2 Single Grid Approximation Subgrid Approximation and Its Performance Preconditioning of the Fine Grid Systems 3 Overlapping DD method Some Numerical Examples When one can use this method ? Examples 13 / 51

Introduction Single Grid Approximation Discretization Subgrid Approximation and Its Performance Preconditioning of the Fine Grid Systems Weak formulation 0 (Ω) n := V and p ∈ L 2 Find u ∈ H 1 0 (Ω) := W such that � � � µ ∇ u : ∇ v + µ (˜ κ u · v ) d x + p ∇ · v d x = f · v d x ∀ v ∈ V � Ω Ω Ω ∀ q ∈ L 2 q ∇ · u d x = 0 0 (Ω) Ω The solution of this problem has unique solution ( u , p ) ∈ V × W . 14 / 51

Introduction Single Grid Approximation Discretization Subgrid Approximation and Its Performance Preconditioning of the Fine Grid Systems The finite element of Brezzi, Douglas, and Marini of degree 1 For this finite element we have: On a rectangle T the polynomial space is characterized by { v = P 2 1 + span { curl ( x 2 1 x 2 ) , curl ( x 1 x 2 2 }} ; with dof normal velocity H pressure ( V H , W H ) ⊂ ( H 0 ( div ; Ω) , L 2 0 (Ω)) := V × W ; Has a natural variant for n = 3. 15 / 51

Introduction Single Grid Approximation Discretization Subgrid Approximation and Its Performance Preconditioning of the Fine Grid Systems DG FEM, Wang and Ye (2007) on a Single Grid Since the tangential derivative along the internal edges will be in general discontinuous, i.e. V H � H 1 0 (Ω) ; therefore we will apply the discontinuous Galerkin method: Find ( u H , p H ) ∈ ( V H , W H ) such that for all ( v H , q H ) ∈ ( V H , W H ) � a ( u H , v H ) + b ( v H , p H ) = F ( v H ) (4) b ( u H , q H ) = 0 . Because of the nonconformity of the FE spaces, the bilinear form a ( u H , v H ) has a special form given below. 16 / 51

Introduction Single Grid Approximation Discretization Subgrid Approximation and Its Performance Preconditioning of the Fine Grid Systems Discretization of Wang and Yang 2007 on Single Grid � τ + b ( v H , p H ) := p H ∇ · v H d x e n − n + Ω e � e T − T + τ − f · v H d x F ( v H ) := e Ω � � µ ∇ u H : ∇ v H + µ κ u H · v H ) d x a ( u H , v H ) := (˜ T T ∈T H � � � � − α − µ ˜ { { u H } } � v H � + { { v H } } � u H � | e | � u H � � v H � ds � �� � e � �� � e ∈E H symmetrization stabilization 2 ( n + e · ∇ ( v · τ + e ) | e + + n − e · ∇ ( v · τ − }| e := 1 { { v } e ) | e − ) � v � | e := v | e + · τ + e + v | e − · τ − e 17 / 51

Introduction Single Grid Approximation Discretization Subgrid Approximation and Its Performance Preconditioning of the Fine Grid Systems Fine and Coarse Grid Spaces: Arbogast (2004) Fine and coarse triangulation T h and T H . FE spaces with ( W h consist of bubbles) W H , h = W h ⊕ W H ⊂ L 2 0 , V H , h = V h ⊕ V H ⊂ H 0 ( div ) ∇ · V h = W h and ∇ · V H = W H 1 Crucial properties: v h · n = 0 on ∂ T , ∀ v h ∈ V h and ∂ T ∈ T H 2 W H ⊥ W h 3 T h T H Ω 18 / 51

Introduction Single Grid Approximation Discretization Subgrid Approximation and Its Performance Preconditioning of the Fine Grid Systems Splitting of the Spaces Unique decomposition in coarse and fine-grid components yields: find u H + u h ∈ W h ⊕ W H , and p H + p h ∈ V h ⊕ V H such that decomposed solution a ( u H + u h , v H + v h ) + b ( v H + v h , p H + p h ) = F ( v H + v h ) , b ( u H + u h , q H + q h ) = 0 , for all v H + v h ∈ W h ⊕ W H and q H + q h ∈ V h ⊕ V H . 19 / 51