ELF Reservoir Test Problem for Numerical Homogenization Methods

Contents:
Map Image of ELF Reservoir
Problem Description
This page describes the ELF Data test problem for homogenization tests. This test problem is relatively simple realistic test for homogenization methods. In this case the configuration is based on a 30 by 30 grid. The test involves the creation of a 6 by 6 homogenized grid for use in simulations of flow through the porous medium.
Problem/Data Links:
The following is a list of raw data sets 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.
Homogenization Results:
The following documents the results for various homogenization methods applied to the reservoir data. All of the methods presented are applied to the 30 by 30 grid to obtain the results given on a 6 by 6 grid.

Arithmetic Average:

* 1 2 3 4 5 6
5
10.0000 0.0 0.0
0.0 10.0000 0.0
0.0 0.0 10.0000
8.41600 0.0 0.0
0.0 8.41600 0.0
0.0 0.0 8.41600
1.68400 0.0 0.0
0.0 1.68400 0.0
0.0 0.0 1.68400
0.10000 0.0 0.0
0.0 0.10000 0.0
0.0 0.0 0.10000
0.64 0.0 0.0
0.0 0.64 0.0
0.0 0.0 0.64
7.84000 0.0 0.0
0.0 7.84000 0.0
0.0 0.0 7.84000
4
5.968 0.0 0.0
0.0 5.968 0.0
0.0 0.0 5.968
9.20800 0.0 0.0
0.0 9.20800 0.0
0.0 0.0 9.20800
0.10000 0.0 0.0
0.0 0.10000 0.0
0.0 0.0 0.10000
0.10000 0.0 0.0
0.0 0.10000 0.0
0.0 0.0 0.10000
0.496 0.0 0.0
0.0 0.496 0.0
0.0 0.0 0.496
7.84000 0.0 0.0
0.0 7.84000 0.0
0.0 0.0 7.84000
3
1.00000 0.0 0.0
0.0 1.00000 0.0
0.0 0.0 1.00000
6.40000 0.0 0.0
0.0 6.40000 0.0
0.0 0.0 6.40000
2.94400 0.0 0.0
0.0 2.94400 0.0
0.0 0.0 2.94400
0.10000 0.0 0.0
0.0 0.10000 0.0
0.0 0.0 0.10000
4.06000 0.0 0.0
0.0 4.06000 0.0
0.0 0.0 4.06000
4.852 0.0 0.0
0.0 4.852 0.0
0.0 0.0 4.852
2
1.00000 0.0 0.0
0.0 1.00000 0.0
0.0 0.0 1.00000
3.19600 0.0 0.0
0.0 3.19600 0.0
0.0 0.0 3.19600
7.84000 0.0 0.0
0.0 7.84000 0.0
0.0 0.0 7.84000
0.748 0.0 0.0
0.0 0.748 0.0
0.0 0.0 0.748
8.41600 0.0 0.0
0.0 8.41600 0.0
0.0 0.0 8.41600
1.68400 0.0 0.0
0.0 1.68400 0.0
0.0 0.0 1.68400
1
0.964 0.0 0.0
0.0 0.964 0.0
0.0 0.0 0.964
0.46 0.0 0.0
0.0 0.46 0.0
0.0 0.0 0.46
9.28000 0.0 0.0
0.0 9.28000 0.0
0.0 0.0 9.28000
10.0000 0.0 0.0
0.0 10.0000 0.0
0.0 0.0 10.0000
8.41600 0.0 0.0
0.0 8.41600 0.0
0.0 0.0 8.41600
0.10000 0.0 0.0
0.0 0.10000 0.0
0.0 0.0 0.10000
0
0.352 0.0 0.0
0.0 0.352 0.0
0.0 0.0 0.352
0.892 0.0 0.0
0.0 0.892 0.0
0.0 0.0 0.892
6.04000 0.0 0.0
0.0 6.04000 0.0
0.0 0.0 6.04000
10.0000 0.0 0.0
0.0 10.0000 0.0
0.0 0.0 10.0000
4.92400 0.0 0.0
0.0 4.92400 0.0
0.0 0.0 4.92400
0.316 0.0 0.0
0.0 0.316 0.0
0.0 0.0 0.316

Geometric Average:

* 1 2 3 4 5 6
5
10.0000 0.0 0.0
0.0 10.0000 0.0
0.0 0.0 10.0000
4.78630 0.0 0.0
0.0 4.78630 0.0
0.0 0.0 4.78630
0.20892 0.0 0.0
0.0 0.20892 0.0
0.0 0.0 0.20892
0.09999 0.0 0.0
0.0 0.09999 0.0
0.0 0.0 0.09999
0.39810 0.0 0.0
0.0 0.39810 0.0
0.0 0.0 0.39810
5.75439 0.0 0.0
0.0 5.75439 0.0
0.0 0.0 5.75439
4
3.01995 0.0 0.0
0.0 3.01995 0.0
0.0 0.0 3.01995
6.91830 0.0 0.0
0.0 6.91830 0.0
0.0 0.0 6.91830
0.09999 0.0 0.0
0.0 0.09999 0.0
0.0 0.0 0.09999
0.09999 0.0 0.0
0.0 0.09999 0.0
0.0 0.0 0.09999
0.12022 0.0 0.0
0.0 0.12022 0.0
0.0 0.0 0.12022
5.75439 0.0 0.0
0.0 5.75439 0.0
0.0 0.0 5.75439
3
1.00000 0.0 0.0
0.0 1.00000 0.0
0.0 0.0 1.00000
3.98107 0.0 0.0
0.0 3.98107 0.0
0.0 0.0 3.98107
0.43651 0.0 0.0
0.0 0.43651 0.0
0.0 0.0 0.43651
0.09999 0.0 0.0
0.0 0.09999 0.0
0.0 0.0 0.09999
0.63095 0.0 0.0
0.0 0.63095 0.0
0.0 0.0 0.63095
0.91201 0.0 0.0
0.0 0.91201 0.0
0.0 0.0 0.91201
2
1.00000 0.0 0.0
0.0 1.00000 0.0
0.0 0.0 1.00000
0.83176 0.0 0.0
0.0 0.83176 0.0
0.0 0.0 0.83176
5.75439 0.0 0.0
0.0 5.75439 0.0
0.0 0.0 5.75439
0.22908 0.0 0.0
0.0 0.22908 0.0
0.0 0.0 0.22908
4.78630 0.0 0.0
0.0 4.78630 0.0
0.0 0.0 4.78630
0.20892 0.0 0.0
0.0 0.20892 0.0
0.0 0.0 0.20892
1
0.91201 0.0 0.0
0.0 0.91201 0.0
0.0 0.0 0.91201
0.25118 0.0 0.0
0.0 0.25118 0.0
0.0 0.0 0.25118
8.31763 0.0 0.0
0.0 8.31763 0.0
0.0 0.0 8.31763
10.0000 0.0 0.0
0.0 10.0000 0.0
0.0 0.0 10.0000
4.78630 0.0 0.0
0.0 4.78630 0.0
0.0 0.0 4.78630
0.09999 0.0 0.0
0.0 0.09999 0.0
0.0 0.0 0.09999
0
0.19054 0.0 0.0
0.0 0.19054 0.0
0.0 0.0 0.19054
0.75857 0.0 0.0
0.0 0.75857 0.0
0.0 0.0 0.75857
3.63078 0.0 0.0
0.0 3.63078 0.0
0.0 0.0 3.63078
10.0000 0.0 0.0
0.0 10.0000 0.0
0.0 0.0 10.0000
2.51188 0.0 0.0
0.0 2.51188 0.0
0.0 0.0 2.51188
0.17378 0.0 0.0
0.0 0.17378 0.0
0.0 0.0 0.17378

HomCode Average:

* 1 2 3 4 5 6
5
10.0000 0.0 0.0
0.0 10.0000 0.0
0.0 0.0 10.0000
6.41324 1.55927 0.0
1.55927 7.50187 0.0
0.0 0.0 8.41600
0.18193 0.01224 0.0
0.01224 0.14238 0.0
0.0 0.0 1.68399
0.09999 0.0 0.0
0.0 0.09999 0.0
0.0 0.0 0.09999
0.46858 -0.0505 0.0
-0.0505 0.46858 0.0
0.0 0.0 0.63999
3.49299 0.0 0.0
0.0 7.49859 0.0
0.0 0.0 7.83999
4
4.01231 0.21817 0.0
0.21817 2.79111 0.0
0.0 0.0 5.96799
8.08686 0.0 0.0
0.0 8.95748 0.0
0.0 0.0 9.20800
0.09999 0.0 0.0
0.0 0.09999 0.0
0.0 0.0 0.09999
0.09999 0.0 0.0
0.0 0.09999 0.0
0.0 0.0 0.09999
0.11060 0.0 0.0
0.0 0.11059 0.0
0.0 0.0 0.496
5.22661 0.24930 0.0
0.24930 7.27582 0.0
0.0 0.0 7.84000
3
1.00000 0.0 0.0
0.0 1.00000 0.0
0.0 0.0 1.00000
2.64402 -0.0917 0.0
-0.0917 5.79168 0.0
0.0 0.0 6.39999
0.24969 -0.0115 0.0
-0.0115 2.41994 0.0
0.0 0.0 2.94400
0.09999 0.0 0.0
0.0 0.09999 0.0
0.0 0.0 0.09999
0.20578 0.01353 0.0
0.01353 3.12207 0.0
0.0 0.0 4.05999
0.22187 -1.0160 0.0
-1.0160 4.39962 0.0
0.0 0.0 4.852
2
1.00000 0.0 0.0
0.0 1.00000 0.0
0.0 0.0 1.00000
0.65602 0.00595 0.0
0.00595 2.66119 0.0
0.0 0.0 3.19599
5.38584 -0.2656 0.0
-0.2656 7.27828 0.0
0.0 0.0 7.84000
0.21669 -0.0031 0.0
-0.0031 0.22097 0.0
0.0 0.0 0.748
6.05117 0.49200 0.0
0.49200 7.49281 0.0
0.0 0.0 8.41600
0.12455 0.0 0.0
0.0 0.26058 0.0
0.0 0.0 1.68399
1
0.94622 0.0 0.0
0.0 0.94622 0.0
0.0 0.0 0.964
0.21960 0.01633 0.0
0.01633 0.30405 0.0
0.0 0.0 0.46
8.47030 0.0 0.0
0.0 9.07874 0.0
0.0 0.0 9.27999
10.0000 0.0 0.0
0.0 10.0000 0.0
0.0 0.0 10.0000
6.05117 0.49200 0.0
0.49200 7.49281 0.0
0.0 0.0 8.41600
0.09999 0.0 0.0
0.0 0.09999 0.0
0.0 0.0 0.09999
0
0.21263 0.00548 0.0
0.00548 0.14857 0.0
0.0 0.0 0.35199
0.72907 0.0 0.0
0.0 0.87337 0.0
0.0 0.0 0.892
2.56893 -0.0862 0.0
-0.0862 5.38868 0.0
0.0 0.0 6.03999
10.0000 0.0 0.0
0.0 10.0000 0.0
0.0 0.0 10.0000
1.72998 0.05438 0.0
0.05438 4.08795 0.0
0.0 0.0 4.92400
0.16397 -0.0116 0.0
-0.0116 0.16397 0.0
0.0 0.0 0.316

Linear Boundary Condition Average:

* 1 2 3 4 5 6
5
10.0000 0.0 0.0
0.0 10.0000 0.0
0.0 0.0 1.00000
7.65059 0.24868 0.0
0.24868 7.77201 0.0
0.0 0.0 1.00000
1.30941 0.42976 0.0
0.42976 0.89483 0.0
0.0 0.0 1.00000
0.09999 0.0 0.0
0.0 0.09999 0.0
0.0 0.0 1.00000
0.53311 -0.1068 0.0
-0.1068 0.53311 0.0
0.0 0.0 1.00000
6.30101 0.19749 0.0
0.19749 7.51528 0.0
0.0 0.0 1.00000
4
4.81801 1.21642 0.0
1.21642 4.40243 0.0
0.0 0.0 1.00000
8.75578 -0.0895 0.0
-0.0895 9.14835 0.0
0.0 0.0 1.00000
0.09999 0.0 0.0
0.0 0.09999 0.0
0.0 0.0 1.00000
0.09999 0.0 0.0
0.0 0.09999 0.0
0.0 0.0 1.00000
0.35283 0.14316 0.0
0.14316 0.35283 0.0
0.0 0.0 1.00000
6.35797 0.49494 0.0
0.49494 7.53469 0.0
0.0 0.0 1.00000
3
1.00000 0.0 0.0
0.0 1.00000 0.0
0.0 0.0 1.00000
4.72628 -0.6322 0.0
-0.6322 5.99691 0.0
0.0 0.0 1.00000
1.40395 -0.4465 0.0
-0.4465 2.61036 0.0
0.0 0.0 1.00000
0.09999 0.0 0.0
0.0 0.09999 0.0
0.0 0.0 1.00000
2.02986 0.79100 0.0
0.79100 3.52055 0.0
0.0 0.0 1.00000
2.80986 0.52198 0.0
0.52198 4.54068 0.0
0.0 0.0 1.00000
2
1.00000 0.0 0.0
0.0 1.00000 0.0
0.0 0.0 1.00000
1.58931 -0.3852 0.0
-0.3852 2.87509 0.0
0.0 0.0 1.00000
6.53032 -0.4973 0.0
-0.4973 7.53575 0.0
0.0 0.0 1.00000
0.51883 0.08331 0.0
0.08331 0.54866 0.0
0.0 0.0 1.00000
7.12504 0.65745 0.0
0.65745 7.96998 0.0
0.0 0.0 1.00000
0.38081 0.19124 0.0
0.19124 1.53635 0.0
0.0 0.0 1.00000
1
0.95964 0.00435 0.0
0.00435 0.95964 0.0
0.0 0.0 1.00000
0.33082 0.08980 0.0
0.08980 0.38264 0.0
0.0 0.0 1.00000
8.93662 -0.0703 0.0
-0.0703 9.23243 0.0
0.0 0.0 1.00000
10.0000 0.0 0.0
0.0 10.0000 0.0
0.0 0.0 1.00000
7.12504 0.65745 0.0
0.65745 7.96998 0.0
0.0 0.0 1.00000
0.09999 0.0 0.0
0.0 0.09999 0.0
0.0 0.0 1.00000
0
0.31757 0.04913 0.0
0.04913 0.21871 0.0
0.0 0.0 1.00000
0.81163 0.00879 0.0
0.00879 0.88721 0.0
0.0 0.0 1.00000
4.57413 -0.6464 0.0
-0.6464 5.60059 0.0
0.0 0.0 1.00000
10.0000 0.0 0.0
0.0 10.0000 0.0
0.0 0.0 1.00000
3.31017 0.71296 0.0
0.71296 4.45884 0.0
0.0 0.0 1.00000
0.25754 -0.0584 0.0
-0.0584 0.25754 0.0
0.0 0.0 1.00000

Harmonic Average:

* 1 2 3 4 5 6
5
10.0000 0.0 0.0
0.0 10.0000 0.0
0.0 0.0 10.0000
0.59382 0.0 0.0
0.0 0.59382 0.0
0.0 0.0 0.59382
0.11881 0.0 0.0
0.0 0.11881 0.0
0.0 0.0 0.11881
0.09999 0.0 0.0
0.0 0.09999 0.0
0.0 0.0 0.09999
0.21739 0.0 0.0
0.0 0.21739 0.0
0.0 0.0 0.21739
3.16455 0.0 0.0
0.0 3.16455 0.0
0.0 0.0 3.16455
4
0.82236 0.0 0.0
0.0 0.82236 0.0
0.0 0.0 0.82236
1.12107 0.0 0.0
0.0 1.12107 0.0
0.0 0.0 1.12107
0.09999 0.0 0.0
0.0 0.09999 0.0
0.0 0.0 0.09999
0.09999 0.0 0.0
0.0 0.09999 0.0
0.0 0.0 0.09999
0.10412 0.0 0.0
0.0 0.10412 0.0
0.0 0.0 0.10412
3.16455 0.0 0.0
0.0 3.16455 0.0
0.0 0.0 3.16455
3
1.00000 0.0 0.0
0.0 1.00000 0.0
0.0 0.0 1.00000
2.17391 0.0 0.0
0.0 2.17391 0.0
0.0 0.0 2.17391
0.15365 0.0 0.0
0.0 0.15365 0.0
0.0 0.0 0.15365
0.09999 0.0 0.0
0.0 0.09999 0.0
0.0 0.0 0.09999
0.16556 0.0 0.0
0.0 0.16556 0.0
0.0 0.0 0.16556
0.19054 0.0 0.0
0.0 0.19054 0.0
0.0 0.0 0.19054
2
1.00000 0.0 0.0
0.0 1.00000 0.0
0.0 0.0 1.00000
0.25075 0.0 0.0
0.0 0.25075 0.0
0.0 0.0 0.25075
3.16455 0.0 0.0
0.0 3.16455 0.0
0.0 0.0 3.16455
0.14116 0.0 0.0
0.0 0.14116 0.0
0.0 0.0 0.14116
0.59382 0.0 0.0
0.0 0.59382 0.0
0.0 0.0 0.59382
0.11881 0.0 0.0
0.0 0.11881 0.0
0.0 0.0 0.11881
1
0.73529 0.0 0.0
0.0 0.73529 0.0
0.0 0.0 0.73529
0.15625 0.0 0.0
0.0 0.15625 0.0
0.0 0.0 0.15625
5.81395 0.0 0.0
0.0 5.81395 0.0
0.0 0.0 5.81395
10.0000 0.0 0.0
0.0 10.0000 0.0
0.0 0.0 10.0000
0.59382 0.0 0.0
0.0 0.59382 0.0
0.0 0.0 0.59382
0.09999 0.0 0.0
0.0 0.09999 0.0
0.0 0.0 0.09999
0
0.13368 0.0 0.0
0.0 0.13368 0.0
0.0 0.0 0.13368
0.48076 0.0 0.0
0.0 0.48076 0.0
0.0 0.0 0.48076
2.01612 0.0 0.0
0.0 2.01612 0.0
0.0 0.0 2.01612
10.0000 0.0 0.0
0.0 10.0000 0.0
0.0 0.0 10.0000
1.03734 0.0 0.0
0.0 1.03734 0.0
0.0 0.0 1.03734
0.12755 0.0 0.0
0.0 0.12755 0.0
0.0 0.0 0.12755

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.
Heterogeneous Simulation Results
The results below show the simulation of flow in the original heterogeneous map provided in the ELF Reservoir data set. These are shown to compare with the results with the simulations performed on homogenized versions of the ELF data.

Elliptic Solution Flux Approximation Flow Approximation

Homogenized Simulation Results
Arithmetic Average Simulation Results

Elliptic Solution Flux Approximation Flow Approximation

Geometric Average Simulation Results

Elliptic Solution Flux Approximation Flow Approximation

HomCode Average Simulation Results

Elliptic Solution Flux Approximation Flow Approximation

Linear Boundary Condition Average Simulation Results

Elliptic Solution Flux Approximation Flow Approximation

Harmonic Average Simulation Results

Elliptic Solution Flux Approximation Flow Approximation

Steps to use JHomogenizer to create and work with the inverted-L cell:
To create the results included on these pages you can use the JHomogenizer tool. The instructions for working with the ELF map and creating homogenized maps are in the following how to file.