Accurate Upscaling and Homogenization of Immiscible Displacement Using Streamline Framework

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Accurate Upscaling and Homogenization of
Immiscible Displacement Using Streamline
Framework

• Theofilos Strinopoulos
• Prof. Thomas Hou
• Nov 30, 2005

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Immiscible Incompressible Two Phase Flow
P is the pressure K is the permeability S is the
saturation, volumetric ratio of oil/total
fluid Restrict to IMPES framework.
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Immiscible Incompressible Two Phase Flow
P is the pressure K is the permeability, has
multiple scales S is the saturation, volumetric
ratio of oil/total fluid Restrict to IMPES
framework.
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Flow Properties - Numerical Considerations
200x200. Flow develops fingers.
50x50. Too much numerical diffusion
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Outline

• Introduce an adaptive frame.
• One phase flow ?(S)1, upscaling the saturation
  
• Two phase flow, upscaling of the pressure
• Future directions

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The Pressure-Streamline Frame
where v0 is the Jacobian. Stream function is
constant along flow lines. Each stream tube
carries the same amount of fluid. Transport
equation becomes one dimensional, non
conservative.
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Advantages in Terms of Adaptivity

• Resolve fast channels
• Adaptive time-stepping across, along streamlines
• No cross-wind diffusion
• Moving mesh along streamlines is easy and
  efficient. Sharp shocks, accurate breakthrough
  times

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Moving Mesh Equations
Introduce a map from a parameter space ? to the
physical domain P(t,?) that satisfies The
monitor function is selected so that grid is
focused near the shock. In the moving coordinate
frame the saturation equation is We solve the
saturation equation using time-splitting. (Li,
Tang, Yang)
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Moving Mesh Computations
S
P
p
?
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Efficiency of Moving Mesh - Linear Flux
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Efficiency of Moving Mesh - Linear Flux
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Efficiency of Moving Mesh - Linear Flux
Resolves shock with fewer points.
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Efficiency of Moving Mesh - Linear Flux
Resolves shock with fewer points. Faster algorithm
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Efficiency of Moving Mesh - Nonlinear Flux
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Efficiency of Moving Mesh - Nonlinear Flux
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Efficiency of Moving Mesh - Nonlinear Flux
Resolves shock with fewer points.
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Efficiency of Moving Mesh - Nonlinear Flux
Resolves shock with fewer points. But the two
algorithms have similar cost.
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Convergence properties of scheme in p,?
Convergence rates are the same. Constant in front
is smaller for scheme in p,?.
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Homogenization of the Saturation Equation
Definition of two scale limit
Ansatz
Lowest order
Asymptotic expansion is not unique. Restrict the
information it contains (Hou and Xin, E).
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Homogenization of the Saturation Equation
Definition of two scale limit
Ansatz
Lowest order
Asymptotic expansion is not unique. Restrict the
information it contains (Hou and Xin, E).
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Structure of the Homogenized Equation
Westhead, Hou, Yang define projection Exact
closure (except for the assumption of
periodicity) But computation of the
projection is difficult.
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Coordinate Transformation
Assume the fast scale structure in p,?
p,? reflects the structure of the
permeability. Implement by a change of
coordinates
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Convergence Theorem
Can prove weak convergence at a rate without
assumptions of periodicity and scale separation.
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Derivation of the two-scale limit equations
For nonlinear flux we generalize the test
functions to two scale Young measures and use the
entropy inequality to obtain convergence. For
the derivation we have used the assumption of
periodicity.
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Convergence estimate for linear flux
Characteristics agree at cell boundaries.
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Convergence estimate for linear flux
Initial condition SIC which is Lifschitz
everywhere except at one point where there is a
jump. Saturation can be written in terms of flow
maps. Now decompose the integration region
into a region near the jump and a region away
from it.
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Convergence estimate for linear flux
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Full Homogenization
For linear flux (Tartar) Solve with a Laplace
and Fourier transform and take the limit to
find Can be extended to Riemann problem with
nonlinear flux. (Efendiev, Popov)
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Efficient Upscaling
Equation for the mean and fluctuations of the
saturation Cell problem has only 1 fast
variable and no projection. Efficient upscaling
by solving the equation for the fluctuations
along coarse streamlines and neglecting high
order terms. (Efendiev, Durlofsky) Macrodispersion
term can be computed as
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Fractional Flow - SPE10 36
400x400 is upscaled to 50x50, upscaling factor of
64
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Saturation profiles - SPE10 36
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Convergence properties - SPE10 36

• Fine solution is averaged over coarse blocks and
  is compared
• to upscaled solution computed on fine grid
  (upscaling error)
• to upscaled solution with moving mesh (total
  error)

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Computational Cost
400x400 is upscaled to 25x25, upscaling factor of
256 Gains in computational cost due to both
adaptivity and upscaling.
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Upscaling the Pressure Equation
To upscale the saturation we need the fine
velocity. MSFEM (Hou, Wu, Efendiev) is a FEM with
basis functions f that satisfy ?n the coarse
blocks. They serve as the downscaling
operator. How to obtain the boundary
condition? With a linear boundary condition bases
contain only local information. (Hou, Efendiev,
Ginting, Ewing) proposed to use a fine initial
pressure computation (one time overhead). Their
computations are more accurate.
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Upscaling the Pressure Equation
We will use an MSFEM method in the p0,?0
frame. The p0,?0 frame follows the structure of
the permeability.
Global information into bases leads to more
accuracy Less numerical diffusion
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Two Phase Flow Algorithm

• Solve for p0,?0 on the fine grid and transform to
  p0,?0.
• Solve for basis functions.
• Solve for the pressure with
• Transform to p,? to solve for the saturation with
• Repeat step 3.
• Transform back to x,y.

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Fractional Flow - 2pt geostatistics
400x400 is upscaled to 50x50, upscaling factor of
64
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Saturation profiles - 2pt geostatistics
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Convergence Properties - 2pt geostatistics
The L2 difference of the initial and final
velocity was 0.204.
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Future Directions

• Accurate interpolation methods
• Stochastic homogenization in flow based frame
• Homogenization outside the IMPES framework
• Homogenization of other equations with a change
  of coordinates
• 1D Pressure equation in stream tubes

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Acknowledgements

• Thomas Hou
• Yalchin Efendiev
• Ruo Li
• Dan Ping Yang
• Victor Ginting