Multiscale Mixed/Mimetic FEM on Complex Geometries Stein Krogstad Jørg E. Aarnes Knut–Andreas Lie SINTEF ICT Oslo, Norway Applied Mathematics 1/18
Outline Motivation and Background 1 Multiscale Mixed FEM 2 Velocity basis functions Subgrid solvers Numerical Example 3 Guidelines for upgridding of complex model Conluding Remarks 4 Applied Mathematics 2/18
Corner-Point Grids Industry standard for modelling complex reservoir geology Specified in terms of: areal 2D mesh of vertical or inclined pillars each volumetric cell is restriced by four pillars each cell is defined by eight corner points, two on each pillar Applied Mathematics 3/18
Motivation Often too much details in geomodels to run reservoir simulations directly ⇒ model coarsening is necessary. Applied Mathematics 4/18
Motivation Often too much details in geomodels to run reservoir simulations directly ⇒ model coarsening is necessary. Standard upscaling Difficult to obtain coarse scale parameters consistently. Need to resample : coarse grid does not match fine grid. Applied Mathematics 4/18
Motivation Often too much details in geomodels to run reservoir simulations directly ⇒ model coarsening is necessary. Standard upscaling Difficult to obtain coarse scale parameters consistently. Need to resample : coarse grid does not match fine grid. Multiscale Mixed FEM (MsMFEM); Incorporates fine scale features in coarse model basis functions. Coarse grid can (in principle) be any partition of the fine grid. Applied Mathematics 4/18
Motivation Often too much details in geomodels to run reservoir simulations directly ⇒ model coarsening is necessary. Standard upscaling Difficult to obtain coarse scale parameters consistently. Need to resample : coarse grid does not match fine grid. Multiscale Mixed FEM (MsMFEM); Incorporates fine scale features in coarse model basis functions. Coarse grid can (in principle) be any partition of the fine grid. Goal: Automated accurate upgridding Applied Mathematics 4/18
Model Equations Total velocity: Elliptic pressure equation: v = − λ ( S ) K ∇ p v = v o + v w ∇ · v = q Total mobility: Hyperbolic saturation equation: λ = λ w ( S ) + λ o ( S ) φ∂S = k rw ( S ) /µ w + k ro ( S ) /µ o ∂t + ∇ · ( vf ( S )) = q w Saturation water: S Fractional flow water: f ( S ) = λ w ( S ) /λ ( S ) Applied Mathematics 5/18
Mixed Methods Weak formulation: × L 2 such that Find ( v, p ) ∈ H 1 , div 0 � � v ∈ H 1 , div ( λ K ) − 1 ˆ v · v dx − p ∇ · ˆ v dx = 0 , ∀ ˆ , 0 � � p ∈ L 2 . p ∇ · v dx = ˆ q ˆ p dx, ∀ ˆ Applied Mathematics 6/18
Mixed Methods Weak formulation: × L 2 such that Find ( v, p ) ∈ H 1 , div 0 � � v ∈ H 1 , div ( λ K ) − 1 ˆ v · v dx − p ∇ · ˆ v dx = 0 , ∀ ˆ , 0 � � p ∈ L 2 . p ∇ · v dx = ˆ q ˆ p dx, ∀ ˆ Multiscale discretization: Seek solutions in low-dimensional subspaces U ms ⊂ H 1 , div and V ⊂ L 2 , 0 where local fine-scale properties are incorporated into the basis functions. Applied Mathematics 6/18
Mixed Methods – Lowest Order Discretization Given finite bases { φ i } = V ⊂ L 2 and { ψ k } = U ⊂ H 1 , div , the 0 resulting linear system reads � B � 0 � � � � C v = , C T O − p q where � � ψ T k ( λ K ) − 1 ψ l dx B kl = and C ki = φ i ∇ · ψ k dx. Applied Mathematics 7/18
Mixed Methods – Lowest Order Discretization Given finite bases { φ i } = V ⊂ L 2 and { ψ k } = U ⊂ H 1 , div , the 0 resulting linear system reads � B � 0 � � � � C v = , C T O − p q where � � ψ T k ( λ K ) − 1 ψ l dx B kl = and C ki = φ i ∇ · ψ k dx. Use hybridization to obtain SPD system Applied Mathematics 7/18
Multiscale Mixed FEM (MsMFEM) Grids and Basis Functions Assume we are given a fine grid with permeability and porosity attached to each fine-grid cell: Applied Mathematics 8/18
Multiscale Mixed FEM (MsMFEM) Grids and Basis Functions Assume we are given a fine grid with permeability and porosity attached to each fine-grid cell: We construct a coarse grid, and choose the discretization spaces U ms and V such that: Applied Mathematics 8/18
Multiscale Mixed FEM (MsMFEM) Grids and Basis Functions Assume we are given a fine grid with permeability and porosity attached to each fine-grid cell: T i We construct a coarse grid, and choose the discretization spaces U ms and V such that: For each coarse block T i , there is a basis function φ i ∈ V . Applied Mathematics 8/18
Multiscale Mixed FEM (MsMFEM) Grids and Basis Functions Assume we are given a fine grid with permeability and porosity attached to each fine-grid cell: T i T j We construct a coarse grid, and choose the discretization spaces U ms and V 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 8/18
Example: A Three-Block Domain − → Three Basis Functions Applied Mathematics 9/18
Multiscale Basis Functions for Velocity Each basis function ψ is the (numerical) solution of a one-phase local flow-problem over two neighboring blocks T i , T j : ψ = − K ∇ φ with � w i ( x ) , for x ∈ T i ∇ · ψ = − w j ( x ) , for x ∈ T j , with BCs ψ · n = 0 on ∂ ( T i ∪ Γ ij ∪ T j ). Applied Mathematics 10/18
Multiscale Basis Functions for Velocity Each basis function ψ is the (numerical) solution of a one-phase local flow-problem over two neighboring blocks T i , T j : ψ = − K ∇ φ with � w i ( x ) , for x ∈ T i ∇ · ψ = − w j ( x ) , for x ∈ T j , with BCs ψ · n = 0 on ∂ ( T i ∪ Γ ij ∪ T j ). Weights w i , w j : Applied Mathematics 10/18
Subgrid Solvers MsMFEM requires that a conservative numerical method is used to compute velocity basis functions. Applied Mathematics 11/18
Subgrid Solvers MsMFEM requires that a conservative numerical method is used to compute velocity basis functions. Alternatives for corner-point grids: Mixed FEM on tetrahedral subgrid of corner-point grid TPFA or MPFA finite-volume methods Mimetic finite-difference methods Applied Mathematics 11/18
Subgrid Solvers MsMFEM requires that a conservative numerical method is used to compute velocity basis functions. Alternatives for corner-point grids: Mixed FEM on tetrahedral subgrid of corner-point grid TPFA or MPFA finite-volume methods Mimetic finite-difference methods All of the above can be recast in mixed form as a discrete approximation of the bilinear form: � u T ( λ K ) − 1 v ≈ � u i M i v i , Ω E i where u i and v i contain the fluxes of u and v over the cell-faces of E i . Applied Mathematics 11/18
Subgrid Solvers MsMFEM requires that a conservative numerical method is used to compute velocity basis functions. Alternatives for corner-point grids: Mixed FEM on tetrahedral subgrid of corner-point grid TPFA or MPFA finite-volume methods Mimetic finite-difference methods All of the above can be recast in mixed form as a discrete approximation of the bilinear form: � u T ( λ K ) − 1 v ≈ � u i M i v i , Ω E i where u i and v i contain the fluxes of u and v over the cell-faces of E i . ⇒ All applicable in the MsMFEM framework. Applied Mathematics 11/18
Numerical Example: A Wavy Depositional Bed (1) 30 × 30 × 100 logically Cartesian. Corner to corner flow Three different perm fields Varying levels of coarsening Applied Mathematics 12/18
Numerical Example: A Wavy Depositional Bed (2) Coarse Partitioning in Index space Applied Mathematics 13/18
Numerical Example: A Wavy Depositional Bed (3) Coarse Partitioning in Index space Relative error in saturation at 0 . 5PVI: Coarse grid Isotropic Anisotropic Heterogeneous 10 × 10 × 10 0 . 026 0 . 143 0 . 094 6 × 6 × 2 0 . 042 0 . 169 0 . 141 3 × 3 × 1 0 . 065 0 . 127 0 . 106 5 × 5 × 10 0 . 060 0 . 138 0 . 142 1 0.9 0.8 0.7 0.6 water cut 0.5 0.4 0.3 fine grid 10x10x10 0.2 6x6x2 0.1 3x3x1 5x5x10 0 0 0.5 1 1.5 PVI Watercut Logically 5 × 5 × 10 Applied Mathematics 14/18
Can we improve results by altering the coarse grid? Potential problems for MsMFEM Applied Mathematics 15/18
Can we improve results by altering the coarse grid? Potential problems for MsMFEM Bidirectional flow over interfaces: Applied Mathematics 15/18
Can we improve results by altering the coarse grid? Potential problems for MsMFEM Bidirectional flow over interfaces: Flow barriers traversing blocks: Applied Mathematics 15/18
Automated Upgridding Guidelines for choosing good grids 1 Minimize bidirectional flow over Flow direction Flow direction Flow direction Flow direction Flow direction Flow direction interfaces: Avoid unnecessary irregularity 6 7 8 (Blocks 6+7 and 3+8) 5 Avoid single neighbors (Block 4) 2 1 3 Ensure faces transverse to major 4 flow (Block 5). 2 Blocks and faces should follow geological layers (Block 3+8) Flow direction Flow direction 3 Blocks should adapt to flow obstacles whenever possible. 6 7 8 4 For efficiency: reduce number of 5 connections 1 3 2 5 Avoid having too many small blocks Applied Mathematics 16/18
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