6. Relation between permeability and pore space structurepore space structure
6.2. Geometrical explanation of permeabilities
6.2.1. Inverted Boolean models
The two series that were used for our experiments are shown in figure 6.5. The open porosities are comparable to the values for the ROMC and ROME structures. The inverted structures show resemblance to porous media made from fused metal beads, wherein, e.g., heat conduction has been studied [130]. An immediately evident difference is that the pore structure which was convex for the ROMC and ROME structures is now concave. The obstacles in the inverted structures also do not have a minimum size like in case of the ROMC and ROME structure where the minimum size is set by the size of the individual grains.
The geometrical and dynamical properties of the EROMC and EROME structures are summarized in Table 6.2. The N that is given in this case has a different meaning than before, it is the number of holes (of elliptical or circular shape) that were punched into the structure. Here the porosity increases with the number of holes. The N for the inverted structures must not be compared with the N for the ROMC and ROME structures. The values are only shown for the sake of completeness.
As a quick intermezzo let us make an important note on the permeability measurement of structures with porosities close to the percolation threshold: The permeability mea-surement at low porosities is not as simple as for higher porosities, because the formation of only a few principal pathways for the fluid also influences the particle trajectories in the in- and outlet part of the sample (see figure A.1 in the appendix). Yet, these tra-jectories are used to calculate the mean flow velocity. Since all tratra-jectories in the inlet have to pass through one of the orifices into the structure, they cannot follow straight
10The values for ˆN can be found in table A.1 in the appendix.
6. Relation between permeability and pore space structure
a b c d e
f g h i j
Figure 6.5.: EROMC and EROME structures generated from Boolean models of randomly placed overlapping monodisperse circles (ROMC) with radiusr = 30µm(a-e) and ellipses with aspect ratio of 8 and a major axis lengtha= 84µm. In contrast to figure 6.1 the fluid phase consists of the individual grains, which results in a very different morphology. 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 φcce ≈ 0.68 for circles or φcee ≈ 0.34 for ellipses with aspect ratio of 8 is reached.
Table 6.2.:Geometrical and dynamical quantifiers for EROMC and EROME structures shown in figure 6.5. Dcand lc are given in units of lattice sites.
φ φo N χo Dc lc kexp/(cl2c) ksim/(cl2c) σ/σ0 EROMC
a 0.851 0.850 8726 -1842 47.97 8 0.304 0.425 0.478
b 0.742 0.700 6174 -1179 35.93 8 0.21 0.132 0.163
c 0.682 0.550 5273 -791 6.05 6.05 0.04 0.034 0.035
d 0.658 0.400 4835 -574 10.72 8 0.0228 0.021 0.026
e 0.668 0.278 5017 -378 5.85 5.85 0.016 0.019 0.0123
EROME
f 0.850 0.850 9095 -8657 27.66 8 0.7416 0.536 0.6388
g 0.701 0.700 5822 -6348 21.54 8 0.186 0.297 0.3779
h 0.551 0.549 3882 -3865 18.29 8 0.0964 0.131 0.1724
i 0.417 0.400 2602 -1793 12.91 8 0.044 0.049 0.0636
j 0.388 0.270 2405 -1406 13.73 8 0.0111 0.017 0.0227
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6.2. Geometrical explanation of permeabilities
0 0.2 0.4 0.6 0.8 1
0 0.2 0.4 0.6 0.8
1 a φcie φcic
φ k//cl2 c
−0.2 0 0.2 0.4 0.6 0.8 1 10−2
10−1 100
b
(φ−φc)/(1−φc) k//cl2 c
Figure 6.6.: (a) Experimentally (closed symbols) and numerically (smaller open symbols) determined permeabilities of EROMC (squares) and EROME (triangles) structures versus porosity. (b) The same permeabilities versus rescaled porosity. Data points of EROMC and EROME structures deviate strongly from a universal curve. Finite-size effects again lead to structures with porosities below the percolation threshold.
lines which means that the measured velocity will depend on the position in the inlet11. These complications make the measurement of the permeability more complicated and more prone to errors, as, in principle, the whole field of view must be measured. We still used the standard approach for the permeability measurement. As a test for the accuracy of this method we tried another method that was not described in chapter 5.
We directly measured the flow velocity at the small orifice of the EROMC structure with φo = 0.278 by determining the local particle velocity field and calibrating it with the help of the reference channel. By also measuring the area of the orifice and multiplying it by the average fluid velocity across that area the permeability was obtained. With the assumption that for a structure with very low porosity and a correspondingly low permeability the whole pressure drops across the porous part of the sample, a value for the permeability can again be calculated with the help of Darcy’s law. This calculated value was in good agreement with the value previously determined with our standard approach.
When the permeabilities of the EROMC and EROME structures are plotted versus the porosity (figure 6.6 (a)), the same general trend as for the ROMC and ROME structures is observed. The permeabilities decrease with decreasing porosity and vanish at the percolation threshold. The percolation thresholds are now, of course, different. The threshold for the EROMC structures isφcic = 1−φcc≈0.68 and φcie = 1−φce ≈0.3412
11In case of a high-porosity structure these velocity values would be practically the same at any position in the sample. Another problem for low-porosity structures is that equation (5.4) is only valid if the individual resistances are well-connected, which is an unjustified assumption that certainly leads to inaccuracies in our measurements which are hard to estimate.
12Since, e.g., the structures ROMC and EROMC are formed by the same algorithm, the EROMC structure starts conducting just when the ROMC stops. Thus,φcic= 1−φcc. This is generally true
6. Relation between permeability and pore space structure
10−2 10−1 100
10−2 10−1
100 a
σ/σ0 k//cl2 c
10−1 100
10−2 10−1 100
b
(1−χo)/N k//cl2 c
Figure 6.7.:(a) Experimentally (closed symbols) and numerically (open symbols) determined permeabilities of EROMC (square) and EROME (triangles) structures vs. formation factor.
The dashed line, which indicates Katz-Thompson law, is in good agreement with our data. (b) The same permeabilities vs. (1−χo)/N. The dashed line is the fit of (6.1) withα= 1.27 that was used for the ROMC and ROME structures. The agreement with the permeabilities of the structures is bad. The green line corresponds to a fit to the data points yielding and exponent αE = 2.05 that agrees much better with the data.
for the EROME structures. The same reasons (higher probability for overlapping) that led to the formation of large clusters in case of the ROME structures, now leads to an easier formation of conducting clusters where the fluid can flow. As a result, the percolation threshold for the EROME is lower than for the EROMC structures.
Rescaling of the porosities (figure 6.6 (b)) again leads to a more universal behavior for both structure types, but the data points show larger deviations from a single curve than the ROMC and ROME structures. The agreement is again better far off the percolation threshold and exhibits stronger scattering of the data points close to the percolation threshold, which are mainly caused by finite-size effects.
Interestingly, the Katz-Thompson law is still able to predict the permeability with astonishing accuracy, as can be seen in figure 6.7 (a). The measured permeabilities all lie very close to the dashed line that represents the value calculated with the Katz-Thompson law. On the other hand, this is expected, because the values for the conduc-tivity are simulated and take into account all the complexities of the pore space13.
Figure 6.7 (b) shows the permeabilities versus (1−χ0)/Nˆ in a log-log plot. Unlike the data for the ROMC and ROME structures, there is no clear collapse onto a master curve and, therefore, a precise power-law behavior for the two structure types cannot be identified. The dashed black line corresponds to the fit that was obtained for the ROMC and ROME structures. For the EROME structures this fit still yields a somewhat
in 2D continuum percolation.
13At low porosities the simulations require finer and finer grids to give accurate results, i.e., the com-putational effort also increases.
88
6.2. Geometrical explanation of permeabilities
a b
Figure 6.8.: Magnified views of small sections of EROMC (a) and EROME (b) structures withφ= 0.85. As a result of the punching out of circles or ellipses both structures show small isolated obstacles (exemplarily indicated by arrows) that will affect the calculated value for ˆN. Such small obstacles do not occur in ROMC and ROME structures.
acceptable agreement with the measured data over a large range of permeabilities14, but fails totally close to the percolation threshold. In case of the EROMC structure, however, there is practically no predictive power left in the fit over the whole range of porosities.
Thus, equation (6.3) with a fixed exponent does not seem to be of universal value for any kind of structure and, clearly, further parameters of the pore space morphology must be considered. This is not too surprising, since the morphology of the inverted structures is completely different. The green dashed line corresponds to a fit to the EROMC and EROME data points which yielded an exponent of αE = 2.05, but even this curve does not reproduce the measurements in a convincing manner. This indicates that the motivation which gave rise to (6.3) might no longer be valid, that is, the calculation of Nˆ from the Minkowski functionals does not give a sound result.
A possible explanation for the stronger deviation as well as for the scattering of the data points and the discrepancy of simulated and measured values is the occurrence of small isolated obstacles in the EROMC and EROME structures (see figure 6.8) which are nonexistent in ROMC and ROME structures. These small obstacles can certainly have a substantial effect on the value of ˆN without significantly affecting the permeability of the structure. Often it is also hard to tell, if these small obstacles are present in the experimental realizations of the porous samples as they can be very small and might also not be fully formed.
In order to give a quantitative analysis of the structural differences of the four structure types, we determined the distributions of obstacle sizes for structures withφo ≈0.85 and φo ≈ 0.28, which are shown in figure 6.9. For both porosities the distributions for the ROMC (black bars) and ROME structures (red bars) both have their maxima around the size of the grains, which is πr2 = 3632px2 corresponding to log(3632) ≈3.56 in the
14A better agreement could certainly be achieved by using another exponent, but the point here is to show that the behavior that was observed for the ROMC and ROME structures is not universal.
6. Relation between permeability and pore space structure
0 1 2 3 4 5 6
100 101 102 103 a
φo≈0.85
log(A)(px2)
p(A)
ROMC ROME EROMC EROME
0 1 2 3 4 5 6
100 101 102 103 b
φo≈0.28
log(A)(px2)
p(A)
Figure 6.9.:Distribution of logarithmized obstacle sizes for the four different structure types forφo ≈0.85 (a) andφo≈0.28 (b). The inverted structures (EROMC, EROME) show a huge number of very small obstacles due to the different formation algorithm, whereas the minimum size for the normal structures is set by the grain size.
distributions. The very few smaller obstacles simply stem from grains lying at the edge of the structure that are cut off. At the low porosity (b) the distributions extend to larger values, for larger clusters are formed by overlapping grains.
For the EROMC (green bars) and EROME structures (blue bars) the distributions look very distinct. For both types there are a very large number of tiny obstacles that only cover an area of a fewpx2. This very large number of small obstacles might not have a big influence on the flow properties of the structure, but it will surely be accounted for in equation (6.3) which is a possible explanation why our formula does not work so well for EROMC and EROME structures.
We can also rationalize the lacking agreement in another way: In section 6.2 we motivated equation (6.1), or more precisely, the term 1−χNo
by interpreting it as an overlapping probability of the individual grains and this could be easily understood for ROMC and ROME structures, where one just adds grain after grain15 until the desired porosity is reached. When we tried to generalize equation (6.1) by the introduction of Nˆ, which was obtained with the help of the Minkowski functionals, namely the area, surface and Euler characteristic of the pore space, we still had an excellent agreement with the measured data, i.e., ˆN obtained from the Minkowski functionals was compa-rable withN. Yet, for the inverted structures the generalized equation did not produce satisfactory results. Thinking again about the motivation for equation (6.1), we see some complications arising with the inverted structures, as now the obstacles are formed by what is left over after punching out the circles or ellipses. Therefore, the question to be answered is: What is the probability for the left-overs to overlap? This question seems odd, because we cannot count something like the individual grains we had before. Even with full knowledge of the formation process we cannot come up with a number, which
15This also means that the number of obstacles one has put into the structure can be counted easily.
90
6.2. Geometrical explanation of permeabilities could also mean that the simple usage of the Minkowski functionals of the pore space is not the way to go. Maybe the solution to the problem with the inverted structure is to find a suitably chosen Boolean model that best replicates the generated structures and then relate their Minkowski functionals to the permeability.