Idealized Fracture Test Problem

Idealized Fracture Test Problem for Numerical Homogenization Methods

Contents:

Problem Description
Problem/Data Links.
Homogenization Results.
Simulation Results.
Steps to use JHomogenizer to work with the idealized fracture.
Back to Test Problems

Map Image of the Idealized Fracture

Left Matrix Region Fracture Zone Right Matrix Region

Problem Description

This page describes an idealized fracture test problem for numerical homogenization tests. This test problem was proposed by Alain Bourgeat as a means of testing and illustrating the behavior of numerical homogenization on maps that have large contrasts in elliptic coefficients. The basic configuration is a narrow fracture zone separating two regions with lower coefficient values. The figures below show the entire domain along with the three separate regions. The entire domain is comprised of a 52 by 50 set of grid blocks with a constant coefficient value in each grid block. The left and right matrix regions are represented by a 25 by 50 set of grid blocks. The fracture zone is represented by a 2 by 50 set of grid blocks.

The goal is to perform homogenization on the entire region or to combine homogenized versions of the subregions to test the effect of homogenization on the fracture.

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.

Original Heterogeneous Maps: These maps include the entire region and each of the original subregions: the left matrix map, the fracture zone map, and the right matrix map.
Entire Region Homogenized Maps: These map data files are defined on a 52 by 50 grid with coefficient value the same in all of the grid blocks. These files do not respect the boundaries of the three heterogeneous regions.
Combined Region Homogenized Maps: These maps are also defined on a 50 by 52 grid. However, each of the separate regions has been extracted, been homogenized, and then incorporated into a map with the three constant coefficient regions reconnected to create an approximation of the map.

Homogenization Results:

The following gives the values of the tensors in the region or subregion for various types of homogenization applied to the map. Results are given for

the entire region,
the left matrix region,
the right matrix region, and
boundary layer regions

where the boundary layer regions are regions that include cells neighboring the fracture zone. For example, the simplest case is to take one of the matrix cells from the left region and one of the cells from the right region and include them with the fracture zone. This allows for a smoother transition between the regions with large differences in the value of the tensor.

Region Homogenized:

Homogenization Method:

Homogenization Tensor:

Entire Region

Arithmetic Average:

41.4974	0.0000	0.0000
0.0000	41.4974	0.0000
0.0000	0.0000	41.4974

Geometric Average:

29.9091	0.0000	0.0000
0.0000	29.9091	0.0000
0.0000	0.0000	29.9091

Harmonic Average:

2.12667	0.0000	0.0000
0.0000	2.12667	0.0000
0.0000	0.0000	2.12667

HomCode Average:

2.47067	0.01341	0.0000
0.01341	40.8541	0.0000
0.0000	0.0000	41.4972

Linear Boundary Condition Average:

3.5035	0.0059	0.0000
0.0059	40.9342	0.0000
0.0000	0.0000	1.0000

Fractured Media Tensor

Not Applicable

Linear Boundary Condition Wavelet Average:

Not Applicable

Periodic Wavelet Average:

Not Applicable

Left Matrix

Arithmetic Average:

3.3827	0.0000	0.0000
0.0000	3.3827	0.0000
0.0000	0.0000	3.3827

Geometric Average:

2.4670	0.0000	0.0000
0.0000	2.4670	0.0000
0.0000	0.0000	2.4670

Harmonic Average:

2.0822	0.0000	0.0000
0.0000	2.0822	0.0000
0.0000	0.0000	2.0822

HomCode Average:

2.58975	0.00898	0.00000
0.00898	2.42125	0.0000
0.0000	0.0000	3.3872

Linear Boundary Condition Average:

2.7398	0.0209	0.00000
0.0209	2.4798	0.0000
0.0000	0.0000	1.0000

Fractured Media Tensor

Not Applicable

Periodic Wavelet

Not Applicable

Linear Boundary Condition Wavelet

Not Applicable

Fracture Zone

Arithmetic Average:

1001.18	0.0000	0.0000
0.0000	1001.18	0.0000
0.0000	0.0000	1001.18

Geometric Average:

1001.18	0.0000	0.0000
0.0000	1001.18	0.0000
0.0000	0.0000	1001.18

Harmonic Average:

1001.18	0.0000	0.0000
0.0000	1001.18	0.0000
0.0000	0.0000	1001.18

HomCode Average:

1001.18	0.0000	0.0000
0.0000	1001.18	0.0000
0.0000	0.0000	1001.18

Linear Boundary Condition Average:

1001.18	0.0000	0.0000
0.0000	1001.18	0.0000
0.0000	0.0000	1.0000

Fractured Media Tensor

Not Applicable

Periodic Wavelet

Not Applicable

Linear Boundary Condition Wavelet

Not Applicable

Right Matrix

Arithmetic Average:

2.6072	0.0000	0.0000
0.0000	2.6072	0.0000
0.0000	0.0000	2.6072

Geometric Average:

2.2112	0.0000	0.0000
0.0000	2.2112	0.0000
0.0000	0.0000	2.2112

Harmonic Average:

1.9821	0.0000	0.0000
0.0000	1.9821	0.0000
0.0000	0.0000	1.9821

HomCode Average:

2.31846	-0.01098	0.0000
-0.01098	2.15021	0.0000
0.0000	0.0000	2.60715

Linear Boundary Condition Average:

2.3555	-0.0198	0.0000
-0.0198	2.1803	0.0000
0.0000	0.0000	1.0000

Fractured Media Tensor

Not Applicable

Periodic Wavelet

Not Applicable

Linear Boundary Condition Wavelet

Not Applicable

Simulation Results:

The following graphics show results from simulations performed on 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 above. The simulation results shown below were obtained using maps generated by the JHomogenizer tool. The graphics below were also generated using the JHomogenizer tool. Contents:

Heterogeneous Simulation Results
Homogenization of Entire Region Simulation Results.
Homogenization of Subregion Simulation Results.
Homogenization with Boundary Layer Simulation Results.

Heterogeneous Simulation Results:

Simulations were performed on the map given above. Our interest is looking at the effect of the homogenization methods on the simulation of fluid flowing through porous media. The simulation of the original heterogeneous map is the following.

Elliptic Solution	Flux Approximation	Flow Approximation (1500 Steps)

The rest of the results show the same simulations as above for the maps resulting from homogenization. The maps are obtained by applying the specified homogenization method to each of the three regions separately and then merging the regions back together. The simulations are all done on the same 52 by 50 mesh one which the original heterogeneous map is defined.

Simulations on the Entire Region

The following simulations were performed on the domain where the heterogeneous map has been replaced by a constant tensor throughout the region. The constant tensor is obtained by the specified averaging method. The simulations were performed on the same grid as the heterogeneous map simulations.

Arithmetic Average on Entire Region

Elliptic Solution Flux Approximation Flow Approximation

Geometric Average on Entire Region

Elliptic Solution Flux Approximation Flow Approximation

HomCode Average on Entire Region

Elliptic Solution Flux Approximation Flow Approximation

Linear Boundary Condition Average on Entire Region

Elliptic Solution Flux Approximation Flow Approximation

Harmonic Average on Entire Region

Elliptic Solution Flux Approximation Flow Approximation

Simulations on the Combined Homogenized Regions

The following simulations were performed on the domain where the heterogeneous map has been replaced by a constant tensor in each of the separate regions - the left matrix, the fracture zone, and the right matrix. The constant tensor is obtained in each of the subregions by the specified averaging method and then the maps for each subregion were connected back together. The simulations were performed on the same grid as the heterogeneous map simulations.

Combined Arithmetic Average

Elliptic Solution Flux Approximation Flow Approximation

Combined Geometric Average

Elliptic Solution Flux Approximation Flow Approximation

Combined HomCode Average

Elliptic Solution Flux Approximation Flow Approximation

Combined Linear Boundary Condition Average

Elliptic Solution Flux Approximation Flow Approximation

Combined Harmonic Average

Elliptic Solution Flux Approximation Flow Approximation

Steps to use JHomogenizer to work with the full fracture map:

To create the results included on these pages you can use the JHomogenizer tool. The instructions for homogenization of the idealized fracture data are in the following how to file.

HowTo Homogenize the Fracture Map File