Multiscale Mixed Finite-Element Methods for Simulation .Multiscale Mixed Finite-Element Methods for

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  • Multiscale Mixed Finite-Element Methods forSimulation of Flow in Highly Heterogeneous

    Porous Media

    KnutAndreas Lie

    SINTEF ICT, Dept. Applied Mathematics

    Fifth China-Norway-Sweden Workshop on ComputationalMathematics, Lund, June 57, 2006

    Applied Mathematics June 2006 1/30

  • Two-Phase Flow in Porous Media

    Pressure equation:

    v = q, v = K(x)(S)p,

    Fluid transport:

    tS + (vf(S)) = (D(S,x)S

    )Applied Mathematics June 2006 2/30

  • Physical Scales in Porous Media Flow

    The scales that impact fluid flow in oil reservoirs range from

    the micrometer scale of pores and pore channels

    via dmm scale of well bores and laminae sediments

    to sedimentary structures that stretch across entire reservoirs.

    Applied Mathematics June 2006 3/30

  • Geological ModelsThe knowledge database in the oil company

    Geomodels consist of geometry androck parameters (permeability K andporosity ):

    K spans many length scales andhas multiscale structure

    maxK/minK 1031010

    Details on all scales impact flow

    Gap between simulation models and geomodels:

    High-resolution geomodels may have 107 109 cellsConventional simulators are capable of about 105 106 cells

    Traditional solution: upscaling of parameters

    Applied Mathematics June 2006 4/30

  • Geological ModelsThe knowledge database in the oil company

    Geomodels consist of geometry androck parameters (permeability K andporosity ):

    K spans many length scales andhas multiscale structure

    maxK/minK 1031010

    Details on all scales impact flow

    Gap between simulation models and geomodels:

    High-resolution geomodels may have 107 109 cellsConventional simulators are capable of about 105 106 cells

    Traditional solution: upscaling of parameters

    Applied Mathematics June 2006 4/30

  • Upscaling the Pressure Equation

    Upscaling: combine cells to derive coarse grid, determine effectivecell properties

    Assume that p satisfies theelliptic PDE:

    (K(x)p

    )= q.

    Upscaling amounts to finding anew field K(x) on a coarser gridsuch that

    (K(x)p

    )= q,

    p p, v v .10 20 30 40 50 60

    20

    40

    60

    80

    100

    120

    140

    160

    180

    200

    220

    2 4 6 8 10

    2

    4

    6

    8

    10

    12

    14

    16

    18

    20

    22

    Here the overbar denotes averaged quantities on a coarse grid.

    Applied Mathematics June 2006 5/30

  • Upscaling the Pressure Equation, contd

    How do we represent fine-scale heterogeneities on a coarse scale?

    Arithmetic, geometric, harmonic, or power averaging

    K =( 1|V |

    VK(x)p dx

    )1/pEquivalent permeabilities ( Kxx = QxLx/Px )

    V

    p=1 p=0

    u=0

    u=0

    V

    p=1

    p=0

    u=0 u=0V

    V

    Lx

    Ly

    Applied Mathematics June 2006 6/30

  • Vision Fast Simulation of Geomodels

    Vision:

    Direct simulation of fluid flow on high-resolution geomodels ofhighly heterogeneous and fractured porous media in 3D.

    Why multiscale methods?

    Small-scale variations in the permeability can have a strong impacton large-scale flow and should be resolved properly. Observation:

    the pressure may be well resolved on a coarse grid

    the fluid transport should be solved on the finest scale possible

    Thus: a multiscale method for the pressure equation should providevelocity fields that can be used to simulate flow on a fine scale

    Applied Mathematics June 2006 7/30

  • Developing an Alternative to Upscaling

    We seek a multiscale methodology that:

    incorporates small-scale effects into the discretisation on acoarse scale

    gives a detailed image of the flow pattern on the fine scale,without having to solve the full fine-scale system

    is robust and flexible with respect to the coarse grid

    is robust and flexible with respect to the fine grid and thefine-grid solver

    is accurate and conservative

    is fast and easy to parallelise

    Applied Mathematics June 2006 8/30

  • From Upscaling to Multiscale Methods

    Standard upscaling:

    Coarse grid blocks:

    Flow problems:

    Multiscale method:

    Coarse grid blocks:

    Flow problems:

    Applied Mathematics June 2006 9/30

  • From Upscaling to Multiscale Methods

    Standard upscaling:

    Coarse grid blocks:

    Flow problems:

    Multiscale method:

    Coarse grid blocks:

    Flow problems:

    Applied Mathematics June 2006 9/30

  • From Upscaling to Multiscale Methods

    Standard upscaling:

    Coarse grid blocks:

    Flow problems:

    Multiscale method:

    Coarse grid blocks:

    Flow problems:

    Applied Mathematics June 2006 9/30

  • From Upscaling to Multiscale Methods

    Standard upscaling:

    Coarse grid blocks:

    Flow problems:

    Multiscale method:

    Coarse grid blocks:

    Flow problems:

    Applied Mathematics June 2006 9/30

  • From Upscaling to Multiscale Methods

    Standard upscaling:

    Coarse grid blocks:

    Flow problems:

    Multiscale method:

    Coarse grid blocks:

    Flow problems:

    Applied Mathematics June 2006 9/30

  • From Upscaling to Multiscale Methods

    Standard upscaling:

    Coarse grid blocks:

    Flow problems:

    Multiscale method:

    Coarse grid blocks:

    Flow problems:

    Applied Mathematics June 2006 9/30

  • From Upscaling to Multiscale Methods

    Standard upscaling:

    Coarse grid blocks:

    Flow problems:

    Multiscale method:

    Coarse grid blocks:

    Flow problems:

    Applied Mathematics June 2006 9/30

  • From Upscaling to Multiscale Methods

    Standard upscaling:

    Coarse grid blocks:

    Flow problems:

    Multiscale method:

    Coarse grid blocks:

    Flow problems:

    Applied Mathematics June 2006 9/30

  • From Upscaling to Multiscale Methods

    Standard upscaling:

    Coarse grid blocks:

    Flow problems:

    Multiscale method:

    Coarse grid blocks:

    Flow problems:

    Applied Mathematics June 2006 9/30

  • Multiscale Mixed Finite ElementsFormulation

    Mixed formulation:

    Find (v, p) H1,div0 L2 such that(K)1u v dx

    p u dx = 0, u H1,div0 ,` v dx =

    q` dx, ` L2.

    Multiscale discretisation:

    Seek solutions in low-dimensional subspaces

    Ums H1,div0 and V L2,

    where local fine-scale properties are incorporated into the basisfunctions.

    Applied Mathematics June 2006 10/30

  • (Multiscale) Mixed Finite ElementsDiscretisation matrices

    (B CCT 0

    )(vp

    )=

    (fg

    ),

    bij =

    i(K)1

    j dx,

    cik =

    k i dx

    Basis k for pressure: equal one in cell k, zero otherwise

    Basis i for velocity:

    1.order RaviartThomas: Multiscale:

    Applied Mathematics June 2006 11/30

  • Multiscale Mixed Finite ElementsGrids and Basis Functions

    We assume we are given a fine grid with permeability and porosityattached to each fine-grid block.

    We construct a coarse grid, and choose the discretisation spaces Vand Ums such that:

    For each coarse block Ti, there is a basis function i V .For each coarse edge ij , there is a basis function ij Ums.

    Applied Mathematics June 2006 12/30

  • Multiscale Mixed Finite ElementsGrids and Basis Functions

    We assume we are given a fine grid with permeability and porosityattached to each fine-grid block.

    We construct a coarse grid, and choose the discretisation spaces Vand Ums such that:

    For each coarse block Ti, there is a basis function i V .For each coarse edge ij , there is a basis function ij Ums.

    Applied Mathematics June 2006 12/30

  • Multiscale Mixed Finite ElementsGrids and Basis Functions

    We assume we are given a fine grid with permeability and porosityattached to each fine-grid block.

    Ti

    We construct a coarse grid, and choose the discretisation spaces Vand Ums such that:

    For each coarse block Ti, there is a basis function i V .

    For each coarse edge ij , there is a basis function ij Ums.

    Applied Mathematics June 2006 12/30

  • Multiscale Mixed Finite ElementsGrids and Basis Functions

    We assume we are given a fine grid with permeability and porosityattached to each fine-grid block.

    TiTj

    We construct a coarse grid, and choose the discretisation spaces Vand Ums such that:

    For each coarse block Ti, there is a basis function i V .For each coarse edge ij , there is a basis function ij Ums.

    Applied Mathematics June 2006 12/30

  • Multiscale Mixed Finite ElementsBasis for the Velocity Field

    For each coarse edge ij , define a basisfunction

    ij : Ti Tj R2

    with unit flux through ij and no flowacross (Ti Tj).

    Homogeneous medium Heterogeneous medium

    We use ij = Kij with

    ij =

    {wi(x), for x Ti,

    wj(x), for x Tj ,

    with boundary conditions ij n = 0 on (Ti Tj).

    Global velocity:

    v =

    ij vijij , where vij are (coarse-scale) coefficients.

    Applied Mathematics June 2006 13/30

  • Multiscale Mixed Finite ElementsBasis for Velocity - the Source Weights.

    If Ti contains a source, i.e.,Tiqdx 6= 0, then

    wi(x) =q(x)

    Tiq() d

    Otherwise we may choose

    wi(x) =1

    |Ti|or to avoid high flow through low-perm regions

    wi(x) =trace(K(x))

    Titrace(K()) d

    The latter is more accurate - even for strong anisotropy.

    Applied Mathematics June 2006 14/30

  • Advantage: Accuracy10th SPE Comp