6. Relation between permeability and pore space structurepore space structure
6.1. Permeabilities of Boolean models
In section 2.4 Boolean models were introduced. We said that they show resemblance to naturally occurring porous media and, hence, are of interest to understand flow proper-ties in such media. For our studies we generated ROMC or ROME structures of circles with a radius r = 30µm and ellipses of aspect ratio of 8 with a major axis length of a = 84µm. The size of the structures is 4000 px×4000 px, which corresponds to 3.5 mm × 3.5 mm in the experimental realizations. The radius of the circles for the ROMC structures was set to r = 34 px corresponding to 30µm and the major axis length to a = 96 px for the ROME structures giving an aspect ratio3 of 8. The two series that were used for our experiments are shown in figure 6.1. Starting at a value of φ = 0.85 on the very left4, the porosities decrease from left to right until they are close to the respective percolation threshold, below which there is no sample-spanning cluster of void space and no fluid can flow through the structure. In figure 6.1 only the open pore space, that is, the part that is accessible by the fluid is shown as white area. Similar open porosities for the ROMC and ROME structures were chosen to make the results comparable. As the porosity comes closer to the percolation threshold, the individual grains start to overlap more frequently, until only a few pathways for the fluid remain. This, of course, has a significant influence on the transport properties of these structures.
Table 6.1 summarizes the geometrical and dynamical properties of the ROMC and
3The major and minor axes of the ellipses were chosen to result in an equal area for ellipses and circles, so that the same number of ellipses covers, without overlapping, the same area as the circles.
4Higher values for the porosities were not studied, because the accuracy of the permeability measure-ment decreases as the porosity goes up. This issue has been discussed in 5.3.1.
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6.1. Permeabilities of Boolean models
a b c d e
f g h i j
Figure 6.1.: Boolean models generated from randomly placed overlapping monodisperse cir-cles (ROMC) with radius r = 30µm (a-e) and ellipses with aspect ration of 8 and a major axis length a= 84µm. On the very left the structures (a,f) have a porosity of aboutφ= 0.85.
From left to right the porosities decrease until the respective percolation threshold of about φcc≈0.32 for circles or φce ≈0.66 for ellipses with aspect ratio of 8 is reached.
ROME structures. The geometrical parameters (φ,φo, χand χo) were determined with the open-source software PAPAYA [124] which uses a marching squares algorithm to minimize discretization errors. The critical pore diameters Dc were determined using the Euclidean distance transform [125], which assigns the shortest distance to the next obstacle to each point of the pore space. Next, starting from the largest value of this Euclidean distance map, thresholds for the highest allowed values are lowered until a continuous path through the structure forms. The threshold value for which the first continuous path is obtained is the critical value Dc. The critical value Dc has been calculated for the two-dimensional structure, but since our structures are three-dimensional, the critical pore diameter is limited by the height of the sample, which is set to 8px. Therefore, if Dc> lc, the critical pore diameter is given by lc.
The experimental realizations of these structures were generated by soft lithography.
The length of the resulting sample was 10−11 mm, the height 8−10µm and the width 3.5 mm. The porous structures that were investigated were contained in the center of these samples and had a size of 3.5 mm×3.5 mm.
The permeabilities of the structures were determined by the constant-head method.
The applied pressure was in the range ∆P = 10−50 Pa. As probe for the fluid flow, an aqueous suspension of polystyrene particles with a diameter of 1.3µm was injected into the samples. The permeability was calculated by measuring the mean particle velocity in the channel containing the structures and in the empty reference channels. Possible local variations in the mean particle velocity were accounted for by measuring at different positions in the channels and averaging over the obtained velocities.
In addition to the experimental values, numerical values for the permeabilities were ob-tained by the massively parallel lattice Boltzmann application network (WALBERLA).
The structures that were used in the experiment were modelled by a lattice of size
6. Relation between permeability and pore space structure
Table 6.1.: Geometrical and dynamical quantifiers for ROMC and ROME structures shown in figure 6.1. Dcand lc are given in units of lattice sites.
φ φo N χo Dc lc kexp/(cl2c) ksim/(cl2c) σ/σ0
ROMC
a 0.850 0.850 754 -520 100.018 8 0.641 0.6696 0.7031
b 0.701 0.700 1632 -724 51.306 8 0.385 0.3807 0.4405
c 0.551 0.549 2704 -635 21.984 8 0.137 0.1575 0.2107
d 0.418 0.401 3968 -395 11.506 8 0.047 0.0414 0.0764
e 0.365 0.298 4592 -220 2.421 2.421 0.0158 0.0273 0.0118 ROME
f 0.854 0.850 771 -352 67.429 8 0.3598 0.3333 0.3787
g 0.751 0.700 1387 -275 41.254 8 0.118 0.1193 0.1501
h 0.684 0.549 1840 -146 6.275 6.275 0.0385 0.0355 0.0392
i 0.639 0.400 2176 -80 6.245 6.245 0.02 0.0174 0.0133
j 0.651 0.266 2064 -45 5.957 5.957 0.00696 0.0112 0.00851
8×4000×4000. A pressure of ∆P = 50 Pa, which is comparable to the pressures used in the experiment was assumed.
For a comparison of the obtained simulated and experimental values of different mea-surements the correct normalization is important. Especially close to φc where lc can become very small it is crucial to not compare the raw permeabilities directly, but to first normalize them, that is, compare k/(clc2) of different structures. As we also recall, the Katz-Thompson law (3.10) stated that this value should be equal to the formation factor, i.e., k/(clc2) =σ/σ0.
In figure 6.2 (a) both the experimentally (blue, red large symbols) and numerical values (small white symbols), which are in good agreement, for the ROMC (squares) and ROME (triangles) structures are plotted versus the porosity. A lower porosity means that there are more obstacles hampering the flow of the fluid and, hence, less space for the fluid flow is available. One can also simply think in terms of boundaries:
More obstacles means more boundaries and since the velocity at the boundary (no-slip condition) is zero, these will slow down the fluid flow. Thus, it is intuitively clear that the permeability decreases with decreasing porosity for both types of structures and vanishes around their respective percolation thresholds. Some of the ROME structures (i,j) have porosities slightly below their percolation threshold. Remember that the percolation threshold φc is only a sharp value in an infinite system. In a finite system, there can be percolation at any porosity different from zero, although the probability decreases substantially once the percolation threshold is crossed.
In figure 6.2 (b) the same permeabilities are plotted versus a rescaled porosity (φ− φc)/(1−φc). This rescaling is motivated by Archie’s law σ/σ0 = (φ−φc)µ/(1−φc), which relates the porosity to the formation factor appearing in the Katz-Thompson law (3.10). The exponent µ typically depends on the porosity and morphology of the pore
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