A Multiscale Mixed Finite-Element Solver for Compressible Black-Oil - PowerPoint PPT Presentation

A Multiscale Mixed Finite-Element Solver for Compressible Black-Oil Flow S. Krogstad, K.-A. Lie , J.R. Natvig, H.M. Nilsen, B. Skaflestad, J.E. Aarnes, SINTEF 2009 SPE Reservoir Simulation Symposium 1 / 1

Multiscale Pressure Solvers Two-level methods for equations: with a near-elliptic behavior with strongly heterogeneous coefficients without scale separations Aim: describe global flow patterns on coarse grid accurately account for fine-scale structures Provide a mechanism to recover approximate fine-scale solutions 2 / 1

The Multiscale Mixed Finite Element (MsMFE) Method The algorithm in a nutshell 1) Generate coarse grid (automatically) 44 927 cells ↓ 148 blocks 9 different coarse blocks 3 / 1

The Multiscale Mixed Finite Element (MsMFE) Method The algorithm in a nutshell 1) Generate coarse grid (automatically) 2) Detect all adjacent blocks 44 927 cells ↓ 148 blocks 9 different coarse blocks 3 / 1

The Multiscale Mixed Finite Element (MsMFE) Method The algorithm in a nutshell 1) Generate coarse grid (automatically) 2) Detect all adjacent blocks 44 927 cells ↓ 148 blocks 9 different coarse blocks 3) Compute basis functions Solve flow problem for all pairs of blocks 3 / 1

The Multiscale Mixed Finite Element (MsMFE) Method The algorithm in a nutshell 1) Generate coarse grid (automatically) 2) Detect all adjacent blocks 44 927 cells ↓ 148 blocks 9 different coarse blocks 3) Compute basis functions 4) Build global solution Solve flow problem for all pairs of blocks Basis functions: building blocks for global solution 3 / 1

The Mixed Finite Element (MsMFE) Method Computation of multiscale basis functions Ω i Ω j Ω ij Each cell Ω i : pressure basis φ i Each face Γ ij : velocity basis ψ ij � ψ ij = − λ K ∇ φ ij   w i ( x ) , x ∈ Ω i  ∇ · � ψ ij = − w j ( x ) , x ∈ Ω j   0 , otherwise 4 / 1

The Mixed Finite Element (MsMFE) Method Computation of multiscale basis functions Homogeneous K : Ω i Ω j Ω ij Each cell Ω i : pressure basis φ i Heterogeneous K : Each face Γ ij : velocity basis ψ ij � ψ ij = − λ K ∇ φ ij   w i ( x ) , x ∈ Ω i  ∇ · � ψ ij = − w j ( x ) , x ∈ Ω j   0 , otherwise 4 / 1

The Mixed Finite Element (MsMFE) Method Interpretation of the weight function The weight function distributes ∇ · v on the coarse blocks: � � ( ∇ · v ) | Ω i = ∇ · ( v ij ψ ij ) = w i v ij j j � � = w i v · n ds = w i ∇ · v dx ∂ Ω i Ω i Different roles: Incompressible flow: ∇ · v = q ∇ · v = q − c t ∂ t p − � Compressible flow: j c j v j · ∇ p 5 / 1

The Mixed Finite Element (MsMFE) Method R Choice of weight function, w i = θ ( x ) / Ω i θ ( x ) dx Incompressible flow: � qdx = 0 , θ ( x ) = trace( K ( x )) Ω i � qdx � = 0 , θ ( x ) = q ( x ) Ω i 6 / 1

The Mixed Finite Element (MsMFE) Method R Choice of weight function, w i = θ ( x ) / Ω i θ ( x ) dx Incompressible flow: � qdx = 0 , θ ( x ) = trace( K ( x )) Ω i � qdx � = 0 , θ ( x ) = q ( x ) Ω i Compressible flow: θ ∝ q : compressibility effects concentrated where q � = 0 θ ∝ K : ∇ · v over/underestimated for high/low K Another choice motivated by physics: ∂p θ ( x ) = φ ( x ) , Motivation: c t ∂t ∝ φ 6 / 1

The Mixed Finite Element (MsMFE) Method Key to effiency: reuse of computations Example: 128 3 grid Computational cost consists of: # operations versus upscaling factor 7 basis functions (fine grid) x 10 8 Basis functions Global system 7 global problem (coarse grid) 6 1.2 Fine scale solution (AMG) O(n ) High efficiency for multiphase flows: 5 Elliptic decomposition 4 3 Reuse basis functions 2 Easy to parallelize 1 0 8x8x8 16x16x16 32x32x32 64x64x64 7 / 1

The Mixed Finite Element (MsMFE) Method Recap from 2007 SPE RSS: million-cell models in minutes 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 0.6 Watercut 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 Fine Time (days) Time (days) 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 25 time steps 0.4 0.4 Reference Reference 0.2 0.2 multiscale + streamlines: MsMFEM MsMFEM Nested Gridding Nested Gridding 0 0 142 sec on a 2.4 GHz PC 0 500 1000 1500 2000 0 500 1000 1500 2000 Time (days) Time (days) upscaling/downscaling, multiscale, fine grid 8 / 1

MsMFE for Complex Grids Challenges posed by grids from real-life models Unstructured grids: (Very) high aspect ratios: 800 × 800 × 0 . 25 m Skewed and degenerate cells: Non-matching cells: 9 / 1

MsMFE for Complex Grids Applicable to general unstructured grids Coarse blocks: (arbitrary) connected collection of cells − → fully automated coarsening strategies Coarse blocks: logically Cartesian in index space 10 / 1

MsMFE for Complex Grids Applicable to general unstructured grids Coarse blocks: (arbitrary) connected collection of cells − → fully automated coarsening strategies Coarse blocks: logically Cartesian in index space 10 / 1

MsMFE for Complex Grids Applicable to general unstructured grids Coarse blocks: (arbitrary) connected collection of cells − → fully automated coarsening strategies 10 / 1

MsMFE for Complex Grids Fine-grid formulation Discretization using a mimetic method (Brezzi et al): u E = λ T E ( p E − π E ) , T E = | E | − 1 N E K E N T E + ˜ T E N E : face normals X E : vector from face to cell centroids T E : arbitrarily such that ˜ ˜ T E X E = 0 Key features: Applicable for general polyhedral cells Non-conforming grids treated as conforming polyhedral Generic implementation for all grid types Monotonicity as for MPFA 11 / 1

MsMFE for Complex Grids Example: single phase, homogeneous K , linear pressure drop Grid TPFA MFDM MsMFEM+TPFA MsMFEM + MFDM 12 / 1

MsMFE for Compressible Black-Oil Models Fine-grid formulation Pressure equation: c∂p u · K − 1 � ∂t + ∇ · � u − ζ� u = q, u = − K λ ∇ p � Time-discretization and linearization: p n ν − p n − 1 u n ν − ζ n u n ν − 1 · K − 1 � u n c ν − 1 + ∇ · � ν − 1 � ν = q ∆ t Hybrid system:       B C D u ν 0 C T − V T P ν − 1 p n − 1 + q  =      − p ν P ν − 1 0 ν − 1 D T 0 0 π ν 0 13 / 1

MsMFE for Compressible Black-Oil Models Coarse-grid formulation       Ψ T B Ψ Ψ T C I Ψ T D J 0 u   T   =  I T P p n  � I T P I − p   C 0 f J T D T Ψ π 0 0 0 Ψ – velocity basis functions Φ – pressure basis functions I – prolongation from blocks to cells J – prolongation from block faces to cell faces � C = Ψ T ( C − V ) I − D λ Φ T P I New feature: fine-scale pressure D λ = diag( λ 0 u f ≈ Ψ u , p f ≈ I p + Φ D λ u , i /λ i ) 14 / 1

MsMFE for Compressible Black-Oil Models Example 1: tracer transport in gas (Lunati&Jenny 2006) constant K lognormal K p (0 , t ) = 1 bar, p ( x, 0) = 10 bar, coarse grid: 5 blocks, fine grid: 100 cells 15 / 1

MsMFE for Compressible Black-Oil Models Example 1: tracer transport in gas (Lunati&Jenny 2006) constant K lognormal K p (0 , t ) = 1 bar, p ( x, 0) = 10 bar, coarse grid: 5 blocks, fine grid: 100 cells Remedy: correction functions (Lunati, Jenny et al; Nordbotten) 15 / 1

MsMFE for Compressible Black-Oil Models Example 1: tracer transport in gas (Lunati&Jenny 2006) Approximate residual equation by � � u = ˆ u i , ˆ p = ˆ p i , ˆ Ω i ⊂ Ω Ω i ⊂ Ω such that u ≈ u ms + ˆ u and p ≈ p ms + ˆ p . Local problems: (ˆ u i , ˆ p i ) solves residual equation locally in � Ω i such that Zero right-hand-side in � Ω i \ Ω i Zero flux BCs on ∂ � Ω i 16 / 1

MsMFE for Compressible Black-Oil Models Example 1: tracer transport in gas (Lunati&Jenny 2006) Non-overlapping correction: pressure flux 17 / 1

MsMFE for Compressible Black-Oil Models Example 1: tracer transport in gas (Lunati&Jenny 2006) Overlapping O ( H/ 2) correction: pressure flux 18 / 1

MsMFE for Compressible Black-Oil Models Example 2: block with a single fault pressure saturation w −5 −5 fine scale 0 0 5 5 10 10 15 15 20 20 25 25 30 30 100 100 80 80 500 500 60 400 60 400 300 300 40 40 200 200 20 20 100 100 0 0 0 0 w −5 −5 0 0 MsMFE 5 5 10 10 15 15 20 20 25 25 30 30 100 100 80 80 500 500 60 400 60 400 40 300 40 300 200 200 20 20 100 100 0 0 0 0 Fine grid: 90 × 10 × 16 cells. Coarse grid: 6 × 2 × 4 blocks. 1000 m 3 /day water injected into compressible oil at 205 bar ( p bh of 200 bar). 19 / 1

Conclusions and Outlook The MsMFE method: is flexible with respect to grids allows automated coarsening requires correction functions for compressible flow Future research: adaptivity of basis/correction functions parallelization error estimation (via VMS framework)? 20 / 1