An upscaling scheme for porous media flows

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An upscaling scheme for porous media flows
Model for multi-phase convection dominated flow
in porous media
Saturation is the ratio of water/oil, fluid
velocity convects the saturation. Introduce
the expansions
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Numerical implementation of the upscaling scheme
Average equation
Fluctuation equation
where
All terms in G are computable without calculating
derivatives of S. Treat y as a parameter. Define
a y-grid, with K x L grid points. Then, the above
system becomes a 1 K x L coupled hyperbolic
system in the independent variables x and t.
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Description of the tests performed
The tests we perform are for the 2-D vertical
cross-sectional model. A smooth initial
saturation is supplied
Red corresponds to water, blue corresponds to
oil Water is injected at the left-hand boundary,
displacing the oil
A velocity field is prescribed
Velocity field is highly heterogeneous, with
rapid variation in the vertical direction. This
is typically the case for porous media flows.
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Description of the tests performed
Compare the results for the upscaling scheme with
well-resolved calculations. Vary the number of
coarse grid points used. Compute norms of the
errors at a fixed time. We use Resolved
calculations are performed using 2048 x 2048
grids. Upscaled calculations performed using N x
N grids, N 16, 32, 64, with fluctuations
computed with 64 x 64 subcell resolution. These
tests are performed for the single-phase case
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Results for the homogenized solution
Compare calculated from the upscaling
scheme with that calculated from the resolved
solution. Convergence is 1st order for the
current scheme
errors at t 0.1
max errors at t 0.1
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Results for the reconstruction of the multiscale
solution
Reconstruct the multiscale solution
Compare with the resolved calculations.
Convergence is 1st order for the current scheme
errors at t 0.1
max errors at t 0.1
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Comparison of the resolved solution and the
multiscale reconstruction
32 x 32 coarse grid, 2048 x 2048 multiscale
reconstruction
Reconstructed solution
Resolved solution
error 0.0138 max error 0.1559