Estimating Permeability from Microscale Porous Structures

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Estimating Permeability from Microscale Porous Structures

This tutorial model demonstrates how to compute porosity and permeability of a sphere packing from a fully resolved microscopic model. These values are then used to model the sphere packing on the macroscopic scale using Darcy’s Law.

Model Definition

The geometry used to calculate the permeability in the microscopic scale is shown in the figure below.

Figure 1: Geometry of the unit cell of a sphere packing.

A body-centered cubic (BCC) lattice is built with spheres of two different diameters. The flow around the spheres is described using the Creeping Flow interface. A pressure difference of 1 Pa is applied between inlet (top boundary) and outlet (bottom boundary). The boundaries that correspond to the solid-fluid interface are defined as walls and the remaining boundaries are symmetry boundaries.

From this set up the porosity and permeability can be calculated. The porosity is defined as the fraction of pore space volume Vfluid to total volume Vtot

To calculate the permeability κ (m2) the following relationship is used

whereas the pressure gradient ∇p is replaced by the pressure difference between inlet and outlet Δp divided by the size of the unit cell L and the velocity vector u is replaced by the outlet velocity uout in flow direction (z-direction), hence

The resulting values are then used to model the sphere packing on the macroscopic scale. The block height is 10 times larger than the height of the unit cell and the base area is 3 times larger.

To verify that a macroscopic approach using homogenization by deploying Darcy’s Law gives accurate values, the model compares the mass flow per unit area (L2).

Results and Discussion

The velocity field in the unit cell is shown in Figure 2.

Figure 2: Velocity in the unit cell.

From the simulation the values for the porosity and permeability are obtained, with ε = 0.49571 and

.

Figure 3 shows the pressure distribution in the macroscopic model.

Figure 3: Pressure field for the macroscopic model.

The mass flow per unit area is about 0.72 kg/(m2·s) in both models.

Porous_Media_Flow_Module/Fluid_Flow/permeability_estimation

Modeling Instructions

From the menu, choose .

In the window, click

.

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In the window, click

.

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In the tree, select .

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In the tree, select .

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Insert the geometry sequence from a file.

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In the toolbar, click and choose .

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Browse to the model’s Application Libraries folder and double-click the file permeability_estimation_geom_sequence.mph.

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In the toolbar, click

.

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button in the toolbar.

Add some parameters to the list, that are used to set up the physics. The list already contains the parameters used to create the geometry and were imported automatically with the geometry sequence.

Global Definitions

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In the window, under click .

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In the window for , locate the section.

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In the table, enter the following settings:


Name	Expression	Value	Description
p_in	1[Pa]	1 Pa	Inlet pressure
rho_f	1000[kg/m^3]	1000 kg/m³	Fluid density
mu_f	1e-3[Pa*s]	0.001 Pa·s	Fluid viscosity

Material 1 (mat1)

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In the window, under right-click and choose .

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In the window for , locate the section.

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In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Density	rho	rho_f	kg/m³	Basic
Dynamic viscosity	mu	mu_f	Pa·s	Basic

Creeping Flow (spf)

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In the window, under right-click and choose .

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In the window for , locate the section.

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From the list, choose .

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Locate the section. In the p0 text field, type p_in.

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Select Boundary 6 only.

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In the toolbar, click

and choose .

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Select Boundary 5 only.

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In the toolbar, click

and choose .

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Select Boundaries 1, 2, 9, and 22 only.

Use the to create a variable for the fluid volume and add variables that calculate porosity and permeability.

Mass Properties 1 (mass1)

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In the window, under right-click and choose .

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Clear all check boxes, except the one for .

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In the window, right-click and choose .

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In the window for , locate the section.

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In the table, enter the following settings:


Name	Expression	Unit	Description
por	mass1.volume/L^3		Porosity
u_out	spf.out1.Mflow/rho_f/L^2	m/s	Outlet velocity
k0	u_outmu_fL/p_in	m²	Permeability

The physics-controlled mesh sets up a boundary layer mesh at the walls which ensures a good resolution of the steep velocity gradients in that region.

In the window, under right-click and choose .

In the toolbar, click

.

Two default plots are created automatically. Modify the velocity plot to obtain Figure 2.

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In the window, expand the node, then click .

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In the window for , locate the section.

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In the text field, type 1.

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Locate the section. From the list, choose .

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Right-click and choose .

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In the window for , locate the section.

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From the list, choose .

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In the text field, type 1.

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Click to expand the section. From the list, choose .

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In the toolbar, click

.

1

In the window, right-click and choose .

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In the window for , locate the section.

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In the text field, type 1.

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Locate the section. From the list, choose .

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From the list, choose .

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Right-click and choose .

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In the window for , locate the section.

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From the list, choose and remove the top, bottom and front boundaries as shown below.

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In the window, right-click and choose .

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Select Boundary 6 only.

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In the window for , locate the section.

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In the text field, type 40.

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Locate the section. Find the subsection. From the list, choose .

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Select the check box.

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In the associated text field, type 80.

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In the toolbar, click

.

Color Expression 1

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Right-click and choose .

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In the window for , locate the section.

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In the text field, type p.

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Locate the section. From the list, choose .

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In the window, under click .

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In the window for , click to expand the section.

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From the list, choose .

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Locate the section. Clear the check box.

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Locate the section. Select the check box.

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In the toolbar, click

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Global Evaluation 1

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In the toolbar, click

.

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In the window for , click in the upper-right corner of the section. From the menu, choose .

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Click in the upper-right corner of the section. From the menu, choose .

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Go to the window.

The permeability is about 3−10m2 and the porosity is 0.49571.

Now, add a new component to set up a Darcy’s Law interface and use these values.

In the window, right-click the root node and choose .

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In the toolbar, click

.

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In the window for , locate the section.

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In the text field, type 3*L.

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In the text field, type 3*L.

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In the text field, type 10*L.

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.

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In the toolbar, click

to open the window.

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Go to the window.

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In the tree, select .

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Find the subsection. In the table, clear the check box for .

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Click in the window toolbar.

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In the toolbar, click

to close the window.

Darcy’s Law (dl)

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In the window, under click .

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In the window for , locate the section.

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From the ρ list, choose . In the associated text field, type rho_f.

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From the μ list, choose . In the associated text field, type mu_f.

Porous Matrix 1

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In the window, click .

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In the window for , locate the section.

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From the εp list, choose . In the associated text field, type 0.49571.

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From the κ list, choose . In the associated text field, type 3.0292E-10.

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In the toolbar, click

and choose .

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Select Boundary 4 only.

The macroscopic model is ten times larger in the direction of the pressure gradient than the unit cell. To compare the results of the unit cell with the results of the macroscopic model set the pressure to 10*p_in.

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In the window for , locate the section.

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In the p0 text field, type 10*p_in.

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In the toolbar, click

and choose .

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Select Boundary 3 only.

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In the toolbar, click

to open the window.

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Go to the window.

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Find the subsection. In the tree, select .

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Disable the Creeping Flow interface for this study.

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Click in the window toolbar.

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In the toolbar, click

to close the window.

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In the window, click .

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In the toolbar, click

.

Again default plots are created. Modify the pressure plot to obtain Figure 3.

Arrow Surface 1

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In the window, right-click and choose .

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In the window for , click to expand the section.

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From the list, choose .

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Locate the section. From the list, choose .

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In the toolbar, click

.

Finally, compare the models using predefined mass flow variables. For the unit cell divide the variable by the area of the unit cell which is L^2 and not the area of the outlet boundary. For the macroscopic model use (3*L)^2. The resulting values should be almost identical.

Global Evaluation 2

1

In the toolbar, click

.

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In the window for , click in the upper-right corner of the section. From the menu, choose .

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Locate the section. In the table, enter the following settings:


Expression	Unit	Description
spf.out1.massFlowRate/L^2	kg/(m^2*s)	Unit mass flow

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Global Evaluation 3

1

In the toolbar, click

.

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In the window for , locate the section.

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From the list, choose .

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Click in the upper-right corner of the section. From the menu, choose .

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Locate the section. In the table, enter the following settings:


Expression	Unit	Description
dl.pr2.Mflow/(3*L)^2	kg/(m^2*s)	Unit mass flow

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Click the button. The results appear in the window.