The flow velocity fields are calculated by the selected flow solver in the three directions by setting up the pressure drop and selecting the computation directions (Boundary Conditions). In FlowDict, the default value corresponds to the virtual pressure drop of 0.02 Pa (0.0002 mbar = 2·10‑7 bar) across the structure in the Z-direction.

The directions X and Y can also be selected, so that the flow is calculated in those directions. In this case, the result file shows the average flow velocities in the three directions calculated for the entered pressure drop. Non-zero values in X- and Y-directions mean that the calculation also gave some flux in the directions perpendicular to the direction of the applied pressure drop. This is due to directional anisotropies of the structures and to the lack of boundaries that could stop the flow in the perpendicular directions in contrast to experiments where boundaries are usually present.

In situations where high velocity occurs, non-Darcy behavior happens, that is important to describe fluid flow in porous media. To identify the beginning of non-Darcy flow, one of the criteria is the Reynolds number (Re) which is defined as

where is the density of the fluid (SI units: kg/m3), u is the velocity of the fluid (m/s), is the dynamic viscosity of the fluid (Pa·s or N·s/m2 or kg/m·s), L is a characteristic length (m). Depending on the different types of porous media and different applications, there are different choices for characteristic length. Four options are provided:

1. Numerical Length: Since permeability is expressed in units of m2, the square root of the permeability can be in length units, (Zheng and Grigg, 2006). This option is used for Stokes by default.
2. Maximum Pore Diameter: The characteristic length can be the diameter of the largest through pore. From structure analysis the maximum pore diameter is found. This option is used for Navier-Stokes by default.
3. Maximum Fiber/Particle Diameter: The characteristic length can also be the largest fiber diameter for fibrous media and the largest particle size for grain structures. From structure analysis, the largest fiber/particle diameter is found.
4. Manually specified: When none of the previous three options fit the users’ definition, the characteristic length can be manually specified. The Specify Characteristics Length / (m) must be entered when this option is chosen.

Observe the changes under the Results – Report subtab in the Result Viewer after clicking Apply. The Reynolds number and the Characteristic Length are updated.

In real measurements of Flow Resistivity, the pressure drop is usually applied in the Z-direction. That is, for the default setting of 0.02 Pa we consider:

Then, the mean velocity in the direction of the pressure drop is measured. The measured pressure drop is divided by the measured mean velocity to obtain the measured flow resistivity.

Formally, the flow resistivity tensor is defined as , where is the fluid viscosity.

Since the fluid flow is computed only in one direction, not all entries of the permeability tensor K are available. Thus, when the Flow Resistivity is selected in FlowDict, the flow resistivity in the Z-direction is approximated by:

If the other directions are also selected, the flow resistivity for them is approximated by dividing the viscosity by the corresponding diagonal entries of the permeability tensor. Thus, flow resistivity is NOT a material property, because it always depends on the fluid viscosity.

The Gurley second or Gurley unit describes the number of seconds required for 100 cubic centimeters (1 deciliter) of air to pass through 1.0 square inch of a given material at a pressure differential of 4.88 inches of water (0.176 psi) (ISO 5636-5:2003).

With these values gathered in the table:

the Gurley value can be obtained from Darcy’s law with:

The Gurley value depends on the permeability and the physical length of the computational domain in the directions of interest.

For the Flow Permeability, the structure’s permeability tensor is found from Darcy’s law:

Here is the averaged velocity vector (averaged flux with i=1 corresponding to a pressure drop in the X-direction, i=2 corresponding to a pressure drop in the Y-direction and i=3 corresponding to a pressure drop in the Z-direction). denotes the fluid viscosity, and

with

is the pressure gradient (or pressure difference) in the ith direction. , , and represent the pressure drop in the corresponding direction in Pa.

are the physical lengths of the computational domain in the directions of interest. is the physical length of a voxel, and NX, NY and NZ are the numbers of voxels in the three coordinate directions (X, Y and Z).

For the linear Stokes EJ, Stokes SimpleFFT, and Stokes LIR flow solvers, this description yields a permeability tensor K that is independent of the applied pressure drop, as well as from the used fluid viscosity. Thus, the permeability is considered a material property.

The Navier-Stokes model for the SimpleFFT and LIR flow solver is a non-linear method. Hence, is not proportional to anymore. In the non-linear case, the computed velocities are typically lower than in the linear case. In this case, the permeability tensor K depends on the pressure drop ad the viscosity, and thus, it is NOT a material property.