Module: Absolute Permeability Tensor Calculation ()

Description:

The Absolute Permeability Tensor Calculation module computes one or several lines of the intrinsic permeability tensor.

Note: see about system requirements and hardware platform availability.

Aknowledgments

This module was developed in collaboration with Dominique Bernard, Research Director at ICMCB-CNRS (Pessac, France).

Theoritical details

Some general elements about absolute permeability are exposed in .

The tensorial problem that is solved in the Absolute Permeability Tensor Calculation is closed by imposing periodic boundary conditions to , and the geometry. It means that no pressure or flow rate can be imposed, in contrast to experiment simulation. What makes the fluid flow is the second member of the second equation (). It represents a kind of force that applies to all the fluid particles.

Physical constraints

The absolute permeability is computed with a single-phase flow. The module takes a labeled image as input, but each label of the image has to be part either of the solid or of the fluid phase: only one solid and one fluid phase are considered for this calculation. The solid phase is impermeable: there is no flow in it.

Computational aspects and results

As the computation can be rather long, it is left to the user to select either one or several lines of the absolute permeability tensor. If less than three directions are selected, the corresponding tensor lines are stored in separate tables of a spreadsheet.

Each line of the spreadsheet represents another computation of the property. The columns contain:

• Geometry file: name of the file on which the computation was done;
• Region of interest: bounding box of the ROI on which the computation was performed;
• kxx, kxy, kxz: components of the absolute permeability tensor line that has been computed, in square micrometer.

If all the three directions are selected, the full tensor is computed and the eigenvalues and vectors are also stored in the same spreadsheet. For each full tensor computation, three lines appear in the spreadsheet with suffix .KTensor.Spreadsheet.

In that case, the columns contain:

• Geometry file: name of the file on which the computation was done on the first line, the two other lines are empty;
• Region of interest: bounding box of the ROI on which the computation was performed on the first line, the two other lines are empty;
• kx, ky, kz: absolute permeability tensor on three lines, one per computation direction, in square micrometer;
• Eigenvalues: eigenvalues sorted in decreasing order;
• Eigenvector x, Eigenvector y, Eigenvector z: eigenvectors in lines, each line vector related to its eigenvalue in the previous column.

Note: the absolute permeability tensor is symmetric, thus reading it by columns or by lines is identical. In the spreadsheet, each line corresponds to one tensor line computation, as if only the checkbox corresponding to this direction was checked.

Units and dimensions

The permeability unit [] is used since 1 darcy, a common unit for permeability, almost equals 1 (1 = 0,9869233 ).

Problems and solutions

What does this error dialog mean?

Figure 1: Example of error dialog at the end of a computation.

This dialog usually appears at the end of a computation and means that something went wrong during the computation. The solver did not reach the convergence target in the indicated number of iterations. There can be mainly two reasons to explain that problem.

First, the number of iterations is not large enough. It can be detected when the performed number of iterations equals the maximal number set as parameter. The maximal limit should be increased in the parameters of the modules. For large data volumes, the default value might be too small.

The other reason is more difficult to identify. When the discretization of the volume is too rough, the solver can start oscillating locally. One value is locally perturbed and each iteration modifies it consequently with respect to its absolute value. This value can be very small but the error is computed on relative variation between two time steps. For example, a value of at iteration number 1 and at iteration 2 varied by between these two iterations. The value is very small, probably negligible for the final result, but the error remains large.

Several solutions can be tested to address these issues:

• Try to remove the non-percolating porosity from the input image. Non-percolating porosity is not involved in the physical property, but computation in it can be rather long to finally end to 0. To remove this porosity, the Axis Connectivity module can be used. The can help to apply this processing.
• Reduce the convergence criterion value, so that the solver iterates longer and reaches a lower error value. Also increase the maximum number of iteration, so that the solver is not bound by that number.
• Try to increase the refining coefficient. This parameter is hidden by default in the XLab modules parameters. To modify it, the Advanced Settings checkbox must be selected, then a slider appears below the list of materials. Setting refining coefficient to 2 means that all voxels is divided by 2 in the three directions of space. On one hand, it multiplies the number of unknowns to compute (probably memory consumption and computation time also) by a factor of 8. On the other hand, it increases the numeric precision and often helps the solver to converge when it could not with a refining coefficient of 1.

Note: The GPU-accelerated version of this module requires a CUDA-enabled GPU. All compatible GPUs are listed here. The current implementation of this module uses only one GPU.

Connections:

Data [required]
The input must be a Label Field.

ROI [optional]
This is a Region Of Interest, meaning that the computation will only take into account the volume contained in the ROI.

Initial Velocity Field [optional]

Initial Pressure Field [optional]
The connected fields are used as initial solution of the problem. It means that the result of a previous computation can be reused to restart this computation from where it ended. It can be useful if the computation was interrupted before the targeted error value was reached. Both fields must be connected, otherwise no initialization will be used. Initializing only velocities or pressures does not allow restarting the computation in an efficient way. The connected fields must have the same dimensions and voxel size as the data set. Otherwise, they will not be used for initializing the solution.

Ports:

Options

This checkbox allows overwriting (checked) / creating new outputs (unchecked) when a computation ends.

Pore Space

This port lists all the materials or labels contained in the connected Label Field. It allows selecting which labels or values must be considered as fluid for the computation. By default, the material named Exterior or the value 0 is selected.

Direction

This option allows the user to define which line of the absolute permeability tensor to compute. Separate computations will be performed for each direction specified. If all the three directions are selected, the full tensor is computed.

Spreadsheet

This port selects where the next result must appear. If "append result" is selected, the next result is appended in the connected spreadsheet to the last created table corresponding to the species transport direction of this computation. If "new spreadsheet" is selected, the result is stored in a new spreadsheet. If no spreadsheet is connected, a new one is created.

Outputs

If one of the check boxes is selected, the corresponding output will appear in the Project View at the end of the computation. The two first check boxes stand for the resulting velocity perturbation and pressure perturbation fields (namely the tensor and vector referenced in the ). Source term or closure variable could be used to mathematically describe them. They are selected by default.

The third check box is checked by default. If the box is checked, a spreadsheet with suffix .KTensor.Error.Spreadsheet is added to the Project View. It contains the estimation of the error (or convergence criterion) at each iteration, for each computation (depending on how many directions are selected).

Note: the perturbation fields are automatically created as output of the module if the computation was aborted or did not converge. The created fields can be connected as inputs of the module, so that the computed values can be used as initial solution of the problem (see above for more details).

hxportgroup Compute Device

Compute Device

This port allows choosing the device for the next computation.

If CPU is selected, the computation will use the number of threads defined in the Preferences Performance. CPU is always available.

If GPU is selected, the port CUDA Device appears. GPU could not be selected for some hardware configuration (see below) and is then insensitive.

Note: The GPU-accelerated version of this module requires a CUDA-enabled GPU. All compatible GPUs are listed here. The current implementation of this module uses only one GPU.

CUDA Device

This port shows a list of all the CUDA devices that can be used for the computation. It is visible only if:

• the graphics driver is recent enough to support CUDA computing;
• the license was found ();
• at least one of the existing device has a compute capability greater than or equal to 2.0.

Each line of this port shows the name of the device, an identifier and the total amount of memory available. These information could help choosing the best device for computation.

Warning

This port is shown only when a message needs to be displayed:

• if an insufficient driver, which is not supporting CUDA, is detected;
• if no CUDA device is detected on the hardware;
• if the necessary license is incorrectly detected;
• if none of the existing devices has a compute capability greater than or equal to 2.0.

Otherwise, this port is not visible.

hxportgroup Advanced Settings

Advanced Settings

This port sets whether the additional options for fine tuning the module are visible (ON) or hidden (OFF). The options that appear when switching ON are considered to be expert options, which should not require modification in most cases.

Refining Coefficient

Only appears when "show advanced settings" from port Options is checked. The refining coefficient can be used to artificially oversample an image, by simply dividing all the voxels by an integer value. This method gives a better precision concerning the evaluation of the unknowns. It can be useful to increase it when the throats in the porous materials are very small (a few voxels wide). It is an integer value, which value must be strictly greater than 0. The slider is limited to 2 to avoid an accidental increase of this value, which would imply a heavy increase of unknown numbers, though a dramatic drop in computation time.

Convergence Criterion

Only appears when "show advanced settings" from port Options is checked. It is assumed that convergence is attained when the time derivatives of the unknowns tend to 0. To evaluate this behavior, the convergence criterion is computed as:

where is the current iteration, is the velocity vector, is the pressure field, is the time step and is the artificial compressibility coefficient (please refer to for more details). Numerically, 0 can never be reached; that is why a precision value must be set to indicate that the convergence is sufficient. This is a floating point value which must be greater than 0 and for which a default value of 10 is suggested.

Warning: the default value is not appropriate for any computation. Depending on the geometry of the sample, it might be mandatory to decrease the precision value drastically.

For convenience, the convergence criterion will often be simply called "error" in the GUI, spreadsheets and Console messages.

Iterations

Only appears when "show advanced settings" from port Options is checked. It is important to know that an explicit resolution is used to solve the Stokes equation system for the permeability computation. There are two numbers to set in this port. The first is the minimum number of iterations to compute. It is used to avoid ending the computation loop too early. It could happen in very specific cases, due to numerical oscillation in the first iterations. These oscillations can sometimes make the time derivative falsely lower than the convergence criterion. The second number refers to the maximum number of iterations to compute. It is used to be sure that the computation loop will end even if the convergence criterion cannot be attained because of numerical approximation problems. Each number can be set to a default value that should be enough for most cases. A computation should not need more than 100 000 iterations, which already is a huge value. The oscillations should not remain beyond 500 iterations.

Expose

This checkbox stands for the refined image. The output that is shown in the Project View is the refined subvolume used for the computation. It means that the output field fits in the ROI if it is defined or has the same dimensions as the input data. It also means that the output field is oversampled by the factor indicated in the refining coefficient option.