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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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