Stratified Medium Test Problem for Numerical Homogenization Methods
Contents:
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Map Image of Stratified Medium
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Problem Description
This page describes the stratified porous medium test problem for homogenization
tests. This test problem is a sanity check for homogenization applied to two
dimensional problems. The resulting homogenized tensor should be a diagonal
tensor. The diagonal component for the coordinate direction parallel to the
stratification should be equal to the arithmetic average. The diagonal component
for the coordinate direction perpendicular to the stratification should be equal
to the harmonic average to the two values in the layers.
Problem/Data Links:
The following is a link to the raw data set that can be downloaded and used as
input for your own homogenization codes for testing, input to reservoir
simulators and other applications. These maps can also be downloaded and used
as input for the JHomogenizer application.
- 2 to 1 change in strata:
- 10 to 1 change in strata:
Homogenization Results:
The following documents the results for various homogenization methods applied
to the stratified porous medium. Any homogenization method should produce the
same results for this test problem. Namely, the arithmetic average parallel to
the stratification and the harmonic average perpendicular to the stratification.
Output values for the two cases are shown below.
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Coefficient Ratio:
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Homogenization Method:
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Homogenization Tensor:
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2:1
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Harmonic Average:
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| 1.3333 |
0.0000 |
0.0000 |
| 0.0000 |
1.3333 |
0.0000 |
| 0.0000 |
0.0000 |
1.3333 |
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Geometric Average:
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| 1.4142 |
0.0000 |
0.0000 |
| 0.0000 |
1.4142 |
0.0000 |
| 0.0000 |
0.0000 |
1.4142 |
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Homcode Average:
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| 1.5000 |
0.0000 |
0.0000 |
| 0.0000 |
1.3333 |
0.0000 |
| 0.0000 |
0.0000 |
1.5000 |
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Arithmetic Average:
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| 1.5000 |
0.0000 |
0.0000 |
| 0.0000 |
1.5000 |
0.0000 |
| 0.0000 |
0.0000 |
1.5000 |
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10:1
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Harmonic Average:
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| 1.8182 |
0.0000 |
0.0000 |
| 0.0000 |
1.8182 |
0.0000 |
| 0.0000 |
0.0000 |
1.8182 |
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Geometric Average:
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| 3.1623 |
0.0000 |
0.0000 |
| 0.0000 |
3.1623 |
0.0000 |
| 0.0000 |
0.0000 |
3.1623 |
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Homcode Average:
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| 5.5000 |
0.0000 |
0.0000 |
| 0.0000 |
1.8182 |
0.0000 |
| 0.0000 |
0.0000 |
5.5000 |
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Arithmetic Average:
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| 5.5000 |
0.0000 |
0.0000 |
| 0.0000 |
5.5000 |
0.0000 |
| 0.0000 |
0.0000 |
5.5000 |
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The plain methods, arithmetic, geometric, and harmonic averages, produce
constant components. The constant values on the diagonal are 1.5, 1.3333, and
1.4142, respectively for the simple averaging methods.
Simulation Results:
The following graphics show results for simulations performed on a heterogeneous
maps and the homogenized maps that result from applying various methods of
averaging. The heterogeneous map used in the generation of the flow simulations
looks like the map below. The simulation results shown below were obtained using
maps generated by the JHomogenizer tool. The graphics below were generated using
the JHomogenizer tool.
Example 1: 2:1 ratio - 32x32 blocks:
Solutions below are shown for the heterogeneous data set and data sets that
result from the homogenization methods.
Heterogeneous Simulation Results
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Elliptic Solution
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Flux Solution
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Flow Solution 1
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Flow Solution 2
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Steps to use JHomogenizer to create and work with the stratified porous medium:
To create the results included on these pages you can use the JHomogenizer tool.
The instructions for creating a stratified porous medium and the homogenization
results are in the following how to file.