Multi-Scale Finite-Volume (MSFV) method for elliptic problems

Subsurface flow simulation

Mark van Kraaij, CASA Seminar Wednesday 13 April 2005

Overview

• Introduction• Flow problem• Solution method (MSFV)• Numerical results• Conclusions

Multi-scale finite-volume method for elliptic problems in subsurface flow simulationP.Jenny, S.H.Lee, H.A. TchelepiJournal of Computational Physics 187, 47-67 (2003)

A multiscale finite element method for elliptic problems in composite materials and porous mediaThomas Y. Hou and Xiao-Huis WuJournal of Computational Physics 134, 169-189 (1997)

Introduction

Flow problem with

different scales

Problem

The level of detail exceeds computational capability

Goal

Obtain the large scale solution accurately and efficiently

without resolving the small scale details

L

Flow problem

Incompressible flow in porous media

mobility

permeability tensor

fluid viscosity

pressure

source term

flux

velocity

outward normal

Solution methods

• Homogenization/Upscaling(First four presentations by Yves, Miguel, Heike and Matthias)

– Periodicity restrictions

– Solving problems with many separate scales is expensive

• Multi-scale approaches(Last two presentations by Nico and Mark)

– Random coefficients on fine grid

– Solving problems with continuous scales is no problem

Multi-scale approaches

• Multi-Scale Finite Element Method– Homogeneous elliptic problems with special oscillatory

boundary conditions on each element

– Small-scale influence captured with basis functions

– Small-scale information brought to large scales through

the coupling of the global stiffness matrix

• Multi-Scale Finite-Volume (MSFV)– Based on ideas from Flux-Continuous Finite Difference

and Finite Element Method

– Allows for computing effective coarse-scale transmissibilities

– Conservative at the coarse and fine scales

– Computationally efficient and well suited for massively parallel computation

Finite-volume formulation

• Partition domain into smaller rectangular volumes , i.e. the coarse grid

•

Challenge

Find a good approximation for the flux at that

captures the small scale information for each volume

• In general the flux is expressed as a linear combination of the pressure values at the coarse grid

with the effective transmissibilities

• By definition, the fluxes are continuous across the interfaces and as a result the finite-volume method is conservative at the coarse grid

• Construct a dual grid by

connecting the mid-points

of four adjacent grid-blocks

• Define four local elliptic problems

• Solutions are the dual basis functions for

Construction of transmissibilities

2

4

1

3

1 2

43

• Pressure field within can be obtained as a function

of the coarse-volume pressure values by super-

position of the dual basis functions

• Compute effective transmissibilities

by assembling integral flux contri-

butions across volume interfaces2

4

1

3

Construction of fine-scale velocity field

• Dual basis functions cannot be used to reconstruct fine-scale velocity field because of– large errors in divergence field

– violation local mass balance

• A second set of local fine-scale basis functions is constructed that is– consistent with fluxes across volume interfaces

– conservative with respect to fine-scale velocity field

• Focus on mass balance in :– Define nine local elliptic problems

with prescribed flux on derivedfrom pressure field (take )

– Solutions are the fine-scale basis functions for

Coarse grid (bold solid lines)

Dual grid (bold dashed lines)

Underlying fine grid (fine dotted lines)

B

D

A

C

1 2 3

7 8 9

4 5 6

• Fine-scale pressure field within can be obtained as

a function of the coarse-volume pressure values by

superposition of the fine-scale basis functions

• Compute conservative fine-scale velocity field from

fine-scale pressure and permeability field

Compute 2nd set of fine-scale basis functions:Solve finite volume problem on coarse grid:Reconstruct fine-scale velocity field in (part of) the domain:Compute transmissibilitiesfrom 1st set of basis functions:

Computational efficiency

# volumes fine grid

# volumes coarse grid

# nodes coarse grid

# time steps

# adjacent coarse volumes to a coarse node

# adjacent coarse volumes to a coarse volume

CPU time to solve linear system with n unknowns

CPU time for one multiplication

Example:

fine grid

coarse grid

Numerical results

Configuration

Injection rate = −1

Production rate = +1

Tracer particles at initial time

Fine solution on 30x30 fine grid MS solution on 5x5 coarse grid

MS solution on a 5x5 coarse grid

(reconstructed fine-scale velocity field not divergence free!)

with random variable equally distributed between 0 and 1

1. Random permeability field

Permeability field

Permeability field

Geostatistically generated permeability field with and

of . Correlation lengths: .

2. Permeability field with isotropic correlation structure

Fine solution on 30x30 fine grid MS solution on 5x5 coarse grid

Geostatistically generated permeability field with and

of . Correlation lengths: .

3. Permeability field with anisotropic correlation structure

Permeability fieldFine solution on 30x30 fine grid MS solution on 5x5 coarse gridFine solution on 30x30 fine grid MS solution on 5x5 coarse grid

Conclusions

• Multi-Scale Finite-Volume (MSFV) method for elliptic problems describing flow in porous media

• Conservative on coarse and fine grid• Transmissibilities account for the fine-scale effects• Parallel computations

Possible extensions– Unstructured grids (oversampling technique)

– Multi-phase flow (saturation)