multi-scale finite-volume (msfv) method for elliptic problems subsurface flow simulation mark van...
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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
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)