Multiscale Mixed/Mimetic FEM on Complex Geometries Stein Krogstad - - PowerPoint PPT Presentation

Multiscale Mixed/Mimetic FEM on Complex Geometries

Stein Krogstad Jørg E. Aarnes Knut–Andreas Lie SINTEF ICT Oslo, Norway

Applied Mathematics 1/18

Outline

1

Motivation and Background

2

Multiscale Mixed FEM Velocity basis functions Subgrid solvers

3

Numerical Example Guidelines for upgridding of complex model

4

Conluding Remarks

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

Elliptic pressure equation: v = −λ(S)K∇p ∇ · v = q Hyperbolic saturation equation: φ∂S ∂t + ∇ · (vf(S)) = qw Total velocity: v = vo + vw Total mobility: λ = λw(S) + λo(S) = krw(S)/µw + kro(S)/µo Saturation water: S Fractional flow water: f(S) = λw(S)/λ(S)

Applied Mathematics 5/18

Mixed Methods

Weak formulation: Find (v, p) ∈ H1,div × L2 such that

• (λK)−1ˆ

v · v dx −

• p∇ · ˆ

v dx = 0, ∀ˆ v ∈ H1,div ,

• ˆ

p∇ · v dx =

• qˆ

p dx, ∀ˆ p ∈ L2.

Applied Mathematics 6/18

Mixed Methods

Weak formulation: Find (v, p) ∈ H1,div × L2 such that

• (λK)−1ˆ

v · v dx −

• p∇ · ˆ

v dx = 0, ∀ˆ v ∈ H1,div ,

• ˆ

p∇ · v dx =

• qˆ

p dx, ∀ˆ p ∈ L2. Multiscale discretization: Seek solutions in low-dimensional subspaces Ums ⊂ H1,div and V ⊂ L2, where local fine-scale properties are incorporated into the basis functions.

Applied Mathematics 6/18

Mixed Methods – Lowest Order Discretization

Given finite bases {φi} = V ⊂ L2 and {ψk} = U ⊂ H1,div , the resulting linear system reads B C CT O v −p

• =

q

• ,

where Bkl =

• ψT

k (λK)−1ψl dx

and Cki =

• φi∇ · ψk dx.

Applied Mathematics 7/18

Mixed Methods – Lowest Order Discretization

Given finite bases {φi} = V ⊂ L2 and {ψk} = U ⊂ H1,div , the resulting linear system reads B C CT O v −p

• =

q

• ,

where Bkl =

• ψT

k (λK)−1ψl dx

and Cki =

• φi∇ · ψk 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 Ums 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 Ums and V such that: For each coarse block Ti, there is a basis function φi ∈ 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 Ums and V such that: For each coarse block Ti, there is a basis function φi ∈ V . For each coarse edge Γij, there is a basis function ψij ∈ Ums.

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 Ti, Tj: ψ = −K∇φ with ∇ · ψ =

• wi(x),

for x ∈ Ti −wj(x), for x ∈ Tj, with BCs ψ · n = 0 on ∂(Ti ∪ Γij ∪ Tj).

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 Ti, Tj: ψ = −K∇φ with ∇ · ψ =

• wi(x),

for x ∈ Ti −wj(x), for x ∈ Tj, with BCs ψ · n = 0 on ∂(Ti ∪ Γij ∪ Tj). Weights wi, wj:

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:

• Ω

uT (λK)−1v ≈

• Ei

uiMivi, where ui and vi contain the fluxes of u and v over the cell-faces of Ei.

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:

• Ω

uT (λK)−1v ≈

• Ei

uiMivi, where ui and vi contain the fluxes of u and v over the cell-faces of

• Ei. ⇒ 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 Logically 5 × 5 × 10

0.5 1 1.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 PVI water cut fine grid 10x10x10 6x6x2 3x3x1 5x5x10

Watercut

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

interfaces:

Avoid unnecessary irregularity (Blocks 6+7 and 3+8) Avoid single neighbors (Block 4) Ensure faces transverse to major flow (Block 5).

2 Blocks and faces should follow

geological layers (Block 3+8)

3 Blocks should adapt to flow

• bstacles whenever possible.

4 For efficiency: reduce number of

connections

5 Avoid having too many small

blocks

1 2 3 4 5 6 7 8

Flow direction Flow direction Flow direction Flow direction Flow direction Flow direction

1 3 2 5 6 7 8

Flow direction Flow direction

Applied Mathematics 16/18

Numerical Example: A Wavy Depositional Bed (4)

General Partitionings

Relative error in saturation at 0.5PVI : Coarse grid Isotropic Anisotropic Heterogeneous Physical 0.134 0.274 0.200 Logical 0.060 0.138 0.142 Constrained 0.057 0.148 0.099 Physical Logical Constrained

Applied Mathematics 17/18

Concluding Remarks and Further Work

Presented a multiscale mixed FEM that efficiently eliminates the need for upscaled properties and resampling on complex geomodels. Suggested guidelines for automated upgridding. Further testing on real field models.

Applied Mathematics 18/18

Concluding Remarks and Further Work

Presented a multiscale mixed FEM that efficiently eliminates the need for upscaled properties and resampling on complex geomodels. Suggested guidelines for automated upgridding. Further testing on real field models. Paper: Multiscale mixed/mimetic methods on corner-point grids. Accepted in Computational Geosciences, Special issue on multiscale methods

Applied Mathematics 18/18