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## Multiscale FiniteVolume Method for Multiphase Flow in Porous Media

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### Simulation for better understanding of oil reservoir/ geological modeling. 7/13/09 ... Small scale velocity felid (e.g distribution of the solute transported by the ... – PowerPoint PPT presentation

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Title: Multiscale FiniteVolume Method for Multiphase Flow in Porous Media

1
• Multi-scale Finite-Volume Method
for Multiphase Flow in Porous Media
• Jyotsna Halwai
• Ferienakademie 2008

2
Outline
• Motivation
• Mathematical modeling
• Numerical modeling
• Efficient Techniques
• MSFV method for porous media
• Summary

3
Motivation
• Simulation for better understanding of oil
reservoir/ geological modeling

4
Motivation
• Different processes occur in different
• parts of the domain and on different
• scale.
• What are looking for in geological
• / reservoir modeling?
• Accuracy of the modeling
• Computational efficiency
• Reduction in memory usage

5
Outline
• Motivation
• Mathematical modeling
• Numerical modeling
• Efficient Techniques
• MSFV for Porous Media
• Summary

6
Mathematical modeling
• Simplified pressure equation
• Incompressible flow
• No capillarity effect
• No gravity effect
• Darcys velocity

where
7
Mathematical modeling
• Concentration equation
• where phase concentration
• where phase concentration of k
component
• total concentration
• fractional flow function

8
Outline
• Motivation
• Mathematical modeling
• Numerical modeling
• Efficient Techniques
• MSFV for Porous Media
• Summary

9
Numerical modeling
• Discretisation techniques for numerical modeling
• Finite Deference Method
• Finite Volume Method
• Finite Element Method

10
Numerical modeling
• Finite Deference Method
• Unknown are located at the grid points
• Approximation of the derivatives can be given by
3 schemes
• Forward difference
• Backward difference
• Central difference

11
Numerical modeling
• Finite Volume Method
• Conservation law is already fulfilled by each
control volume and it automatically implies to
the global mass conservation.

Face centered CVs
Node centered CVs
12
Numerical modeling
• Finite Volume Method
• Net flux thru the CV boundary is the sum of
integral over four (2D) or six (3D) faces.

13
Numerical modeling
• Finite Element Method
• Solution region comprises of many small,
interconnected, sub region or elements
• Allows variety of shape / basis functions.
• Gives piece-wise approximation to the governing
equations
• Weak formulation reduces the PDEs to linear or
non-linear system of equations.

14
Outline
• Motivation
• Mathematical modeling
• Numerical modeling
• Efficient Techniques
• MSFV method for porous media
• Summary

15
Efficient Techniques
• Up-scaling

Fine mesh large memory time
requirement Size 1 m X 1 m X 1 m
Coarse mesh less accurate less computation
time Size 50 m X 50 m X 50 m
16
Efficient Techniques
• Down-scaling

Fine mesh large memory time
requirement Size 1 m X 1 m X 1 m
Coarse mesh less accurate less computation
time Size 50 m X 50 m X 50 m
17
Efficient Techniques
• Multi-scaling Up-scaling Downscaling
• Flux across the coarse volume interface can be
computed by multi-scale transmissibilitities.
• Pressure equation can be solved at the coarse
grid.
• Small scale velocity felid (e.g distribution of
the solute transported by the fluid) can be
captured at the fine grid.

18
Outline
• Motivation
• Already learnt
• Numerical modeling
• Efficient Techniques
• MSFV method for Porous Media
• Summary

19
MSFV method for porous media
• Complete description
• Simplified flow problem
• Multi-scale finite volume method
• Finite volume method
• Construction of the effective transmissibilities
• Reconstruction of conservative fine-scale
velocity field
• Implementation of MSFV method

20
MSFV for porous media
• Simplified flow problem elliptic equation
• Assumptions
• Incompressible flow
• No capillarity effect
• No gravity effect

21
MSFV for porous media
• Finite volume method for given flow problem
• Cell centered finite volume method
• Domain divided into smaller volume
• _
• v unit normal vector
• Flux is expressed as
• Fluxes are continuous across the interfaces and
as a result finite volume method is conservative

22
MSFV for porous media
• Construction of the effective transmissibilities

Need to solve 4 elliptic problems and then
pressure is calculated by linear combination of
dual basis functions
23
MSFV for porous media
• Reconstruction of conservative fine-scale
velocity field

24
MSFV for porous media
• Implementation of MSFV method-
• Computation of transmissibilities for coarse
fluxes.
• Construction of fine scale basis functions
• Computation of the coarse solution at new time
level
• Reconstruction of the fine-scale velocity field
in regions of interest
• Solution of the transport equations

25
MSFV method
• Over all features
• Targets the full problem with original resolution
• Maximize the local operation by implementing
boundary conditions at the coarse grid boundary
• Fits into finite volume framework
• Allows for computing effective coarse grid
transmissibility,
• Treat the permeability tensor properly
• Conservative at coarse and fine scales.
• Computational efficient and suitable for
massively parallel computation.

26
Outline
• Motivation
• Already learnt
• Numerical modeling
• Efficient Techniques
• MSFV for Porous Media
• Summary

27
Summary
• Will be given by one of the audience, as
promised. ?
• Can we achieve our goals?
• YES..
• Increased computation efficiency by solving
pressure at coarse grid
• Considerable about of accuracy achieved using
fine scale velocity field computations.
• Reduced memory usage at the course grid

28
References
• Multi-scale finite volume method for compressible
multiphase flow in porous media
• Ivan Lunati, Patrick Jenney
• Institute of Fluid Dynamics, ETH-Zurich,
Switzerland
• Multi-scale finite volume method for elliptic
problems in subsurface flow simulation
• P.Jenney, S.H. Lee, H.A. Tchelepi
• Exploration Production Technology Company
• San Ramon, CA,USA
• An adaptive local-global multi-scale finite
volume element method for two phase flow
simulation
• L. Durlofsky, Y. Efendiev, V. Giniting
• Department of Petroleum Engineering
• Stanford University, CA, USA

29
• Thank you
• for your attention!
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