**3. Fluid flow permeability analysis**

Permeability is a tensor valued measure of the ability of a porous material to transmit fluids. It is defined for slow, steady-state, isothermal, Newtonian fluid flow through a porous medium by Darcys's law (Darcy, 1856; Bear, 1972)

$$
\vec{q} = \frac{\vec{\mathbb{Z}}}{\mu} \mathbf{\mathbb{X}} \times \mathbf{\mathbb{N}} \boldsymbol{\mu}, \tag{1}
$$

where *q* is the superficial volume flux vector and is the dynamic viscosity of the fluid. The pietsometric head is defined by equation *p g* where *<sup>p</sup>* is the pressure and is the density of the fluid, and *g* is the acceleration due to a body force. In general, permeability is a symmetric second-order tensor (Liakopoulos, 1965):

$$\mathbf{x} = \begin{bmatrix} k\_{\chi\chi} & k\_{\chi y} & k\_{\chi z} \\ k\_{y\chi} & k\_{yy} & k\_{yz} \\ k\_{z\chi} & k\_{zy} & k\_{zz} \end{bmatrix} \tag{2}$$

Several theoretical results for permeability coefficients have been reported in the literature. Perhaps the most common formula which can be derived analytically for simplified capillary model is the Kozeny-Carman relation

$$k = \frac{1}{c\tau^2 S\_0^2} \frac{\phi^3}{(1-\phi)^2} \tag{3}$$

The Effect of Tomography Imaging Artefacts on

in the sample holder (b).

coefficient is thus given by

law

place.

Structural Analysis and Numerical Permeability Simulations 475

Fig. 3. Schematic illustration of the measurement set-up (a) and a sandstone sample attached

In the experimental method, the diagonal values of the permeability tensor in the case dependent coordinate system (see e.g. Fig. 1) are found using the integrated form of Darcy's

*<sup>i</sup> Q L <sup>k</sup> P L AP P i out in*

where *Q* is the volumetric flow rate through the sample, *A* is the cross-sectional flow area, *PP P out in* is the pressure drop and *Li* is the length over which the pressure drop takes

Experiments were conducted with air flow in order to prevent structural changes due to swelling of sample material. This is important in order to obtain similar structure of materials in experiments and in numerical flow solution based on the pore geometry given by tomographic images of dry material samples. Equation (5) is valid for the incompressible fluid flows. For gas flows through porous medium, Darcy's law must be slightly modified to account for compressibility effects. For isothermal compressible flow the permeability

/ ()

*q P Q L <sup>P</sup> i out out <sup>k</sup> P L P AP P P i ave out in ave*

where *Pout* is the pressure, *Q* is the volumetric flow rate at the downstream side of the

In the experimental approach, the values of the permeability coefficient in z-direction ( *zz k* ) were calculated using Darcy's law for compressible fluid flow by Eq. (6). The coordinate conventions for the sample types are shown in Fig. 1. Measurements were repeated for five macroscopically identical samples in order to obtain an estimate of the statistical uncertainty

*q*

*ii*

*ii*

medium, and 1 / 2( ) *P P PP ave out in out* (Bear, 1972; Leskelä & Simula, 1998).

of the results. The statistical uncertainty of the experimental results was 20%.

/( )

*i*

*i*

(6)

(5)

where *c* is a constant that depends on the cross section of the capillaries (being 2 *c* for circular cross section), 0 *<sup>S</sup>* is the specific surface area of the sample, is the tortuosity of the flow and is the porosity of the media (Bear, 1972; Dullien, 1979).

### **3.1 Numerical method**

Numerical permeability analyses were done directly on the voxel model of the samples. The diagonal elements of permeability tensor (in Eq. (2)) were obtained within FDM by first solving over a periodic REV the following boundary value problem arising from the homogenisation process:

$$\begin{aligned} \, \_0\nabla \vec{\nabla} \vec{v} - \nabla \delta p - \nabla \, \_0\nabla &= 0, \qquad &\text{in } \Omega\_f\\ \nabla \cdot \vec{v} &= 0 &\text{in } \Omega\_{f'}\\ \vec{v} &= 0 &\text{on } \Gamma \end{aligned} \tag{4}$$

where *f* and represent the fluid volume and the fluid-solid interface, respectively. These three equations are the conservation of momentum, conservation of mass and the noslip condition on the fluid-solid interface. Here *v* stands for the periodic microscopic velocity field, is the pietsometric head and *p* is the first order periodic fluctuation of the pressure *p* .

The boundary value problem Eq. (4) was solved by using FDM implemented in GeoDict2010 R1 (64 bit Linux) software. In the method, the velocity *v* and the pressure *p* are discretised on a staggered grid: velocities are defined on their respective voxel faces and the pressure is defined in the centre of the voxel. Then the partial differential equations are solved by using the FFF-Stokes solver based on Fast Fourier Transform. This solver appears to be fast and memory efficient for large computations dedicated to 3D images (Wiegmann, 2007). Finally, the permeability coefficients of the samples can be deduced from equations (1) and (2) by calculating the volume average of the velocity field over the REV ( *<sup>v</sup> <sup>q</sup>* ) for a given macroscopic gradient of pressure .

During the permeability analysis, periodic boundary condition was enforced on all the sample faces. For tomographic samples, ten extra fluid layers were added in flow direction on both sides of the sample volume. This was done in order to mimic the experimental measurement conditions, see Koivu et al., (2009a) and Koivu et al., (2009b).

#### **3.2 Experimental method**

Permeability coefficients *zz k* of the samples were measured experimentally using the permeability measurement device (PMD) presented in refs (Koivu et al., 2009a; Koivu et al., 2009b). The PMD can be utilised for measuring permeability of porous materials using both liquids and gases as permeating fluids.

For the purpose of this work, the PMD was modified such that the sample was compressed only at its peripheral part to prevent any flow on that region, and the central part was left fully open for flow; see a schematic illustration of the measurement set-up in Fig. 3a. The measured sample size for the wool fibre web and the packaging board had 90 mm diameter. For the measurements, the sand stone samples with diameter of 35 mm and thickness 10 mm was attached with silicone glue into a special sample holder, see Fig. 3b.

where *c* is a constant that depends on the cross section of the capillaries (being 2 *c* for circular cross section), 0 *<sup>S</sup>* is the specific surface area of the sample, is the tortuosity of the

Numerical permeability analyses were done directly on the voxel model of the samples. The

solving over a periodic REV the following boundary value problem arising from the

<sup>2</sup> 0, in

 

 

*µv p <sup>f</sup> <sup>v</sup> <sup>f</sup>*

0 in 0 on

where *f* and represent the fluid volume and the fluid-solid interface, respectively. These three equations are the conservation of momentum, conservation of mass and the no-

The boundary value problem Eq. (4) was solved by using FDM implemented in

*p* are discretised on a staggered grid: velocities are defined on their respective voxel faces and the pressure is defined in the centre of the voxel. Then the partial differential equations are solved by using the FFF-Stokes solver based on Fast Fourier Transform. This solver appears to be fast and memory efficient for large computations dedicated to 3D images (Wiegmann, 2007). Finally, the permeability coefficients of the samples can be deduced from equations (1) and (2) by calculating the volume average of the velocity field over the REV

During the permeability analysis, periodic boundary condition was enforced on all the sample faces. For tomographic samples, ten extra fluid layers were added in flow direction on both sides of the sample volume. This was done in order to mimic the experimental

Permeability coefficients *zz k* of the samples were measured experimentally using the permeability measurement device (PMD) presented in refs (Koivu et al., 2009a; Koivu et al., 2009b). The PMD can be utilised for measuring permeability of porous materials using both

For the purpose of this work, the PMD was modified such that the sample was compressed only at its peripheral part to prevent any flow on that region, and the central part was left fully open for flow; see a schematic illustration of the measurement set-up in Fig. 3a. The measured sample size for the wool fibre web and the packaging board had 90 mm diameter. For the measurements, the sand stone samples with diameter of 35 mm and thickness 10

.

(in Eq. (2)) were obtained within FDM by first

, (4)

and the pressure

stands for the periodic microscopic

*p* is the first order periodic fluctuation of

is the porosity of the media (Bear, 1972; Dullien, 1979).

 

is the pietsometric head and

GeoDict2010 R1 (64 bit Linux) software. In the method, the velocity *v*

measurement conditions, see Koivu et al., (2009a) and Koivu et al., (2009b).

mm was attached with silicone glue into a special sample holder, see Fig. 3b.

flow and

**3.1 Numerical method** 

homogenisation process:

velocity field,

the pressure *p* .

**3.2 Experimental method** 

liquids and gases as permeating fluids.

diagonal elements of permeability tensor

*v*

slip condition on the fluid-solid interface. Here *v*

( *<sup>v</sup> <sup>q</sup>* ) for a given macroscopic gradient of pressure

Fig. 3. Schematic illustration of the measurement set-up (a) and a sandstone sample attached in the sample holder (b).

In the experimental method, the diagonal values of the permeability tensor in the case dependent coordinate system (see e.g. Fig. 1) are found using the integrated form of Darcy's law

$$k\_{ii} = -\frac{\mu \eta\_{\dot{I}}}{\Delta P \;/\ \Delta L\_{\dot{I}}} = \frac{\mu Q \Delta L\_{\dot{i}}}{A (P\_{out} - P\_{\dot{i}m})} \tag{5}$$

where *Q* is the volumetric flow rate through the sample, *A* is the cross-sectional flow area, *PP P out in* is the pressure drop and *Li* is the length over which the pressure drop takes place.

Experiments were conducted with air flow in order to prevent structural changes due to swelling of sample material. This is important in order to obtain similar structure of materials in experiments and in numerical flow solution based on the pore geometry given by tomographic images of dry material samples. Equation (5) is valid for the incompressible fluid flows. For gas flows through porous medium, Darcy's law must be slightly modified to account for compressibility effects. For isothermal compressible flow the permeability coefficient is thus given by

$$k\_{ii} = -\frac{\mu q\_{\dot{i}}}{\Delta P \;/\ \Delta L\_{\dot{i}}} \frac{P\_{out\text{f}}}{P\_{ave}} = \frac{\mu Q \Delta L\_{\dot{i}}}{A(P\_{out\text{f}} - P\_{in\text{f}})} \frac{P\_{out\text{f}}}{P\_{ave}} \tag{6}$$

where *Pout* is the pressure, *Q* is the volumetric flow rate at the downstream side of the medium, and 1 / 2( ) *P P PP ave out in out* (Bear, 1972; Leskelä & Simula, 1998).

In the experimental approach, the values of the permeability coefficient in z-direction ( *zz k* ) were calculated using Darcy's law for compressible fluid flow by Eq. (6). The coordinate conventions for the sample types are shown in Fig. 1. Measurements were repeated for five macroscopically identical samples in order to obtain an estimate of the statistical uncertainty of the results. The statistical uncertainty of the experimental results was 20%.

The Effect of Tomography Imaging Artefacts on

(Drummont & Tahir, 1984).

see Eq. (3).

array of cylinders.

Structural Analysis and Numerical Permeability Simulations 477

Numerically analysed permeability values and corresponding (noise free) analytical value by Drummont & Tahir (1984) are shown in Fig. 5. According to the results, even a small

Fig. 5. Numerically solved permeability values for the hexagonal array of cylinders as a function of noise level. For comparison, analytical value for noise free geometry is also given

Fig. 6. Dimensionless specific surface area as a function of noise level for the hexagonal

The effect of noise on specific surface area was evaluated as a function of noise level, see Fig. 6. The specific surface areas were analysed utilising the marching cubes algorithm (Lorensen & Clive, 1987a; Lorensen & Clive, 1987b). Increase in the amount of noise increases the specific surface area of the simulation geometry and thus decreases the permeability value,

amount of noise has a drastic influence on fluid flow permeability.
