Multiscale methods for modeling flow in porous media : approaching industrial applications Jørg E. Aarnes, Stein Krogstad and Knut–Andreas Lie 1.5 1 0.5 0 20 15 20 15 10 10 5 5 0 0 Applied Mathematics 1/26
Outline How can we promote multiscale methods to the oil-industry 1 Gap between academic research and industry needs Industry-standard geological models 2 The corner-point format Coarse grid-generation Multiscale mixed finite element method 3 Coarse grid formulation Subgrid discretization Why consider multiscale methods 4 Applied Mathematics 2/26
The gap between academia and industry Grids Industry models: Academic models: Simple domains Structured grids Conforming grids Applied Mathematics 3/26
The gap between academia and industry Physics (Flow in porous media) Industry: Academia: Compressible Incompressible Miscible Immiscible Yes! Gravity? Yes. Capillary forces? Pseudofunctions Relative permeability??? etc., etc., etc., ... etc. Applied Mathematics 4/26
The gap between academia and industry What is important Academia: Industry: Can it be published? Money Nice plots! Risk Accuracy Can it handle our models Efficiency Efficiency Practical importance Robustness Applied Mathematics 5/26
Conjecture: MsMFEM has the following key features Based on experience with synthetic Cartesian petroleum reservoir models 100 120 140 160 180 200 220 20 40 60 80 0 0 Accurate: flow scenarios match closely fine grid simulations. 50 Mass conservative: conserves mass 100 120 140 160 180 200 220 20 40 60 80 0 0 on coarse and fine grids. Efficient: basis functions can be 50 computed in parallel and need not 100 120 140 160 180 200 220 20 40 60 80 0 be recomputed. 0 Flexible: unstructured and irregular 50 coarse grids are handled easily. 100 120 140 160 180 200 220 20 40 60 80 0 0 Robust: suitable for models with highly oscillatory coefficients and large grid-cell aspect ratios. 50 Applied Mathematics 6/26
Promoting multiscale methods to the industry Possible scenarios Do you want an amazing multiscale method? Applied Mathematics 7/26
Promoting multiscale methods to the industry Possible scenarios Do you want an amazing multiscale method? A1: Multiscale method??? Applied Mathematics 7/26
Promoting multiscale methods to the industry Possible scenarios Do you want an amazing multiscale method? A2: But, multiscale methods are new and very complex, right? Applied Mathematics 7/26
Promoting multiscale methods to the industry Possible scenarios Do you want an amazing multiscale method? A3: Have they been tested on realistic models? Applied Mathematics 7/26
Promoting multiscale methods to the industry Possible scenarios Do you want an amazing multiscale method? A4: Yes, when it is implemented in my favourite software! Applied Mathematics 7/26
Promoting multiscale methods to the industry Possible scenarios Promoting multiscale methods to the industry is a challenge, but academia must make the first move! Applied Mathematics 7/26
Bridging the gap between academia and industry A prerequisite for conducting simulation studies on full-scale real-field petroleum reservoir models is the ability to handle grids on a corner-point format. Model: corner-point grid without fractures and faults. Physics: incompressible and immiscible two-phase flow, neglecting effects from (gravity and) capillary forces. Applied Mathematics 8/26
Bridging the gap between academia and industry A prerequisite for conducting simulation studies on full-scale real-field petroleum reservoir models is the ability to handle grids on a corner-point format. Model: corner-point grid without fractures and faults. Physics: incompressible and immiscible two-phase flow, neglecting effects from (gravity and) capillary forces. Next: MsMFEM on corner-point grid geological models Applied Mathematics 8/26
Corner-point grids The industry standard for reservoir modeling and simulation The corner-point, or pillar grid format, has become the industry standard for reservoir modeling and simulation. In a corner-point grid the grid-cell corner-points lie on pillars (lines) that extend from the top to the bottom of the reservoir. Applied Mathematics 9/26
Corner-point grids, cont. The data structure for corner-point grids is logically Cartesian, i.e., 1 the pillars are ordered in a logical Cartesian manner, and 2 each layer extends throughout the entire reservoir. Layers may collapse to a hyperplane in certain regions. Collapsed cells are labeled non-active. Active cells have polyhedral shape with 5 – 8 corners. Applied Mathematics 10/26
Corner-point grids, cont. The data structure for corner-point grids is logically Cartesian, i.e., 1 the pillars are ordered in a logical Cartesian manner, and 2 each layer extends throughout the entire reservoir. Layers may collapse to a hyperplane in certain regions. Collapsed cells are labeled non-active. Active cells have polyhedral shape with 5 – 8 corners. In physical space, corner-point grids are unstructured! Applied Mathematics 10/26
Corner-point grids, cont. Examples of degenerate hexahedral cells in corner-point grids Applied Mathematics 11/26
Generating a coarse grid for MsMFEM Let K = { K } be a coarse grid with blocks of “arbitrary” shape, and denote by T = { T } a fine subgrid of K Applied Mathematics 12/26
Grid coarsening procedures In order to avoid resampling of geological data, we assume that grid blocks consists of a union of cells in the fine grid. Applied Mathematics 13/26
Grid coarsening procedures In order to avoid resampling of geological data, we assume that grid blocks consists of a union of cells in the fine grid. Partitioning in physical space: Blocks have approximately equal volume :=) Interfaces become very irregular :=( Applied Mathematics 13/26
Grid coarsening procedures In order to avoid resampling of geological data, we assume that grid blocks consists of a union of cells in the fine grid. Partitioning in physical space: Blocks have approximately equal volume :=) Interfaces become very irregular :=( Partitioning in index space: Block volumes differ significantly, and blocks are irregular :=( Interfaces are usually smooth :=) Applied Mathematics 13/26
Grid coarsening procedures In order to avoid resampling of geological data, we assume that grid blocks consists of a union of cells in the fine grid. Partitioning in physical space: Blocks have approximately equal volume :=) Interfaces become very irregular :=( Partitioning in index space: Block volumes differ significantly, and blocks are irregular :=( Interfaces are usually smooth :=) Volume constrained partitioning in index space: Blocks are irregular and number of neighbors increases :=( Blocks have smooth faces, and approximately equal volume :=) Applied Mathematics 13/26
Multiscale mixed finite element method Examples of grid blocks that arise when partitioning in index space Disconnected blocks are split into a family of connected subblocks. Applied Mathematics 14/26
Multiscale mixed finite element method Model problem Let Ω denote a computational domain, and consider the following model problem v = − k ∇ p, ∇ · v = in Ω , q v · n = 0 on ∂ Ω . Here k is a symmetric and positive definite tensor with uniform upper and lower bounds in Ω. We will refer to p as pressure and v as velocity. Applied Mathematics 15/26
Multiscale mixed finite element method The mixed formulation In mixed FEMs one seeks v ∈ V and p ∈ U such that � � k − 1 v · u dx − p ∇ · u dx = 0 ∀ u ∈ V, Ω Ω � � l ∇ · v dx = ∀ l ∈ U. ql dx Ω Ω Here V ⊂ { v ∈ ( L 2 ) d : ∇ · v ∈ L 2 } and U ⊂ L 2 . Applied Mathematics 16/26
Multiscale mixed finite element method The mixed formulation In mixed FEMs one seeks v ∈ V and p ∈ U such that � � k − 1 v · u dx − p ∇ · u dx = 0 ∀ u ∈ V, Ω Ω � � l ∇ · v dx = ∀ l ∈ U. ql dx Ω Ω Here V ⊂ { v ∈ ( L 2 ) d : ∇ · v ∈ L 2 } and U ⊂ L 2 . In multiscale mixed FEMs the approximation space for velocity is designed so that it embodies the impact of fine scale structures. Applied Mathematics 16/26
Multiscale mixed finite element method Pressure basis functions Associate a basis function χ m for pressure with each grid block: � 1 if x ∈ K m , U = span { χ m : K m ∈ K} where χ m = 0 else. Applied Mathematics 17/26
Multiscale mixed finite element method Velocity basis functions Construct a velocity basis function for each interface ∂K i ∩ ∂K j : V = span { ψ ij } where ψ ij = − k ∇ φ ij and φ ij is determined by no-flow boundary conditions on ( ∂K i ∪ ∂K j ) \ ( ∂K i ∩ ∂K j ), and � q ( K i ) in K i , ∇ · ψ ij = − q ( K j ) in K j , where | k | � if K f dx = 0 , R K | k | q ( K ) = f � if K f dx � = 0 . R K f Applied Mathematics 18/26
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