An Overview of the Multiscale Mixed Finite-Element Method SINTEF ICT, Department of Applied Mathematics Multiscale Workshop, Dr. Holms, Geilo, Dec 5, 2008 Applied Mathematics 05/12/2008 1/63
Multiscale Pressure Solvers Efficient flow solution on complex grids – without upscaling Basic idea: Upscaling and downscaling in one step Pressure on coarse grid (subresolution near wells) Velocity with subgrid resolution everywhere Example: Layer 36 from SPE 10 Applied Mathematics 05/12/2008 2/63
Multiscale Pressure Solvers Two main contenders... Multiscale mixed finite elements Multiscale finite volumes Developed by SINTEF Developed by Jenny/Lee/Tchelepi/.. Main focus on complex grids Focus on flow physics Corner-point grids in 3D Gravity and capillarity Triangular/nonuniform/PEBI Black-oil Automated coarsening Compressibility Complex wells + Stokes–Brinkman, wells, black-oil Only for Cartesian grids, so far. Applications: history match, optimization Applied Mathematics 05/12/2008 3/63
Geological Models as Direct Input to Simulation Complex reservoir geometries Challenges: Industry-standard grids are often nonconforming and contain skewed and degenerate cells There is a trend towards unstructured grids Standard discretization methods produce wrong results on skewed and rough cells The combination of high aspect and anisotropy ratios can give very large condition numbers Corner point: Tetrahedral: PEBI: Applied Mathematics 05/12/2008 4/63
The MsMFE Method in a Nutshell From upscaling to multiscale methods Standard upscaling: ⇓ Coarse grid blocks: ⇓ Flow problems: Applied Mathematics 05/12/2008 5/63
The MsMFE Method in a Nutshell From upscaling to multiscale methods Standard upscaling: ⇓ Coarse grid blocks: ⇓ ⇑ Flow problems: Applied Mathematics 05/12/2008 5/63
The MsMFE Method in a Nutshell From upscaling to multiscale methods Standard upscaling: ⇓ ⇑ Coarse grid blocks: ⇓ ⇑ Flow problems: Applied Mathematics 05/12/2008 5/63
The MsMFE Method in a Nutshell From upscaling to multiscale methods Standard upscaling: ⇓ ⇑ Coarse grid blocks: ⇓ ⇑ Flow problems: Applied Mathematics 05/12/2008 5/63
The MsMFE Method in a Nutshell From upscaling to multiscale methods Standard upscaling: Multiscale method: ⇓ ⇓ ⇑ Coarse grid blocks: Coarse grid blocks: ⇓ ⇑ ⇓ Flow problems: Flow problems: Applied Mathematics 05/12/2008 5/63
The MsMFE Method in a Nutshell From upscaling to multiscale methods Standard upscaling: Multiscale method: ⇓ ⇓ ⇑ Coarse grid blocks: Coarse grid blocks: ⇓ ⇑ ⇓ ⇑ Flow problems: Flow problems: Applied Mathematics 05/12/2008 5/63
The MsMFE Method in a Nutshell From upscaling to multiscale methods Standard upscaling: Multiscale method: ⇓ ⇑ ⇓ ⇑ Coarse grid blocks: Coarse grid blocks: ⇓ ⇑ ⇓ ⇑ Flow problems: Flow problems: Applied Mathematics 05/12/2008 5/63
The MsMFE Method in a Nutshell Mixed formulation for incompressible flow Mixed formulation: × L 2 such that Find ( v, p ) ∈ H 1 , div 0 � � ∀ u ∈ H 1 , div ( λK ) − 1 u · v dx − p ∇ · u dx = 0 , , 0 � � ∀ ℓ ∈ L 2 . ℓ ∇ · v dx = qℓ dx, Multiscale discretization: Seek solutions in low-dimensional subspaces in which local fine-scale properties are incorporated into the basis functions Applied Mathematics 05/12/2008 6/63
The MsMFE Method in a Nutshell Linear system and basis functions Discretisation matrices: � � − 1 ψ j dx, � B � b ij = ψ i λK � � v � � f � C Ω = , C T 0 p g � c ik = φ k ∇ · ψ i dx Ω Raviart–Thomas: Multiscale basis function: Applied Mathematics 05/12/2008 7/63
The MsMFE Method in a Nutshell Grids and basis functions We assume we are given a fine grid with permeability and porosity attached to each fine-grid block. Applied Mathematics 05/12/2008 8/63
The MsMFE Method in a Nutshell Grids and basis functions We assume we are given a fine grid with permeability and porosity attached to each fine-grid block. We construct a coarse grid, and choose the discretisation spaces V and U ms such that: Applied Mathematics 05/12/2008 8/63
The MsMFE Method in a Nutshell Grids and basis functions We assume we are given a fine grid with permeability and porosity attached to each fine-grid block. T i We construct a coarse grid, and choose the discretisation spaces V and U ms such that: For each coarse block T i , there is a basis function φ i ∈ V . Applied Mathematics 05/12/2008 8/63
The MsMFE Method in a Nutshell Grids and basis functions We assume we are given a fine grid with permeability and porosity attached to each fine-grid block. T i T j We construct a coarse grid, and choose the discretisation spaces V and U ms such that: For each coarse block T i , there is a basis function φ i ∈ V . For each coarse edge Γ ij , there is a basis function ψ ij ∈ U ms . Applied Mathematics 05/12/2008 8/63
The MsMFE Method in a Nutshell Local flow problems For each coarse edge Γ ij , define a basis function with unit flux through Γ ij and no flow across ∂ ( T i ∪ T j ). Local flow problem: � w i ( x ) , for x ∈ T i , ψ ij = − λK ∇ φ ij , ∇ · ψ ij = − w j ( x ) , for x ∈ T j , with boundary conditions ψ ij · n = 0 on ∂ ( T i ∪ T j ). Global velocity: v = � ij v ij ψ ij , where v ij are (coarse-scale) coefficients. Applied Mathematics 05/12/2008 9/63
The MsMFE Method in a Nutshell Automated generation of coarse grids The MsMFE method allows fully automated coarse gridding strategies: grid blocks need to be connected, but can have arbitrary shapes Corner-point grids: the coarse blocks are logically Cartesian in index space Applied Mathematics 05/12/2008 10/63
The MsMFE Method in a Nutshell Workflow with automated upgridding in 3D 1) Coarsen grid by uniform partitioning in 2) Detect all adjacent blocks index space for corner-point grids 44 927 cells ↓ 148 blocks 9 different coarse blocks 3) Compute basis functions 4) Block in coarse grid: component for building global solution ( w i ( x ) , ∇· ψ ij = − w j ( x ) , for all pairs of blocks Applied Mathematics 05/12/2008 11/63
The MsMFE Method in a Nutshell Computational efficiency: order-of-magnitude argument, 128 × 128 × 128 grid Multigrid more efficient when computing pressure once. Why bother with multiscale pressure solvers? 7 Full simulation: O (10 2 ) time x 10 8 Computation of basis functions Solution of global system steps. 7 Basis functions need not be 6 Fine scale solution recomputed 5 4 Also: 3 Possible to solve very large 2 problems 1 Easy parallelization 0 8x8x8 16x16x16 32x32x32 64x64x64 Applied Mathematics 05/12/2008 12/63
The MsMFE Method in a Nutshell Example: 10 th SPE Comparative Solution Project SPE 10, Model 2: Water-cut curves at the four producers Producer A Producer B 1 1 Producer A 0.8 0.8 Injector Producer D Watercut Watercut 0.6 0.6 Tarbert 0.4 0.4 Upper Ness Producer B Reference Reference 0.2 0.2 MsMFEM MsMFEM Nested Gridding Nested Gridding 0 0 0 500 1000 1500 2000 0 500 1000 1500 2000 Producer C Time (days) Time (days) Fine grid: 60 × 220 × 85 Producer C Producer D 1 1 Coarse grid: 5 × 11 × 17 0.8 0.8 2000 days production Watercut 0.6 Watercut 0.6 0.4 0.4 4M + streamlines: Reference Reference 0.2 0.2 MsMFEM MsMFEM Nested Gridding Nested Gridding 2 min 22 sec on 2.4 GHz 0 0 0 500 1000 1500 2000 0 500 1000 1500 2000 Time (days) Time (days) desktop PC upscaling/downscaling, 4M/streamlines, fine grid Applied Mathematics 05/12/2008 13/63
Implementation Details for MsMFE There are certain choices.... Choice of weighting function in definition of basis functions Boundary conditions (overlap and global information) Assembly of linear system Fine-grid discretization Generation of coarse grids Applied Mathematics 05/12/2008 14/63
Implementation Details for MsMFE Choice of the weight function Interpretation of the weight function: � � ( ∇ · v ) | T i = w i ∇ · ( v ij ψ ij ) = w i v ij j j � � = w i v · nds = w i ∇ · v ∂ Ti T i That is, w i distributes ∇ · v among the cells in the coarse grid Different roles: Incompressible flow: ∇ · v = q Compressible flow: ∇ · v = q − c t ∂ t p − � j c j v j · ∇ p Applied Mathematics 05/12/2008 15/63
Implementation Details for MsMFE Weight function: incompressible flow For incompressible flow, we have that � � 0 , if T i qdx = 0 , � � ( ∇ · v ) | T i = w i v ij , v ij = � T i qdx, otherwise j j Thus � qdx = 0 ⇒ ∇ · v = 0 , ∀ w i > 0 T i q � qdx � = 0 ⇒ ∇ · v = q, if w i = � T i qdx T i Applied Mathematics 05/12/2008 16/63
Implementation Details for MsMFE Choice of weight function: uniform Uniform source: 1 w i ( x ) = | T i | v h Ω Ω k 1 2 h p =0 p =1 T � T ij j i v Ω Ω l 3 4 k l low ( k l ) and high ( k h ) permeability streamlines from basis function Applied Mathematics 05/12/2008 17/63
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