refer-8. Conclusion and outlook
ence channel, a proportionality factor between particle and fluid velocity can be obtained.
This calibration circumvents the problem of the size-dependent velocity of the particles and allows for a calculation of the mean flow velocity. The calibration factor was used to measure the permeability of the investigated porous structures, which consisted of a number N of randomly placed overlapping circles (ROMC) or ellipses (ROME). More-over, we measured the permeabilities of structures where the conducting and obstacle phase have been exchanged resulting in structures of conducting circles (EROMC) and ellipses (EROME) that have a very different pore space geometry. The measured perme-abilities were complemented by permeperme-abilities obtained by lattice Boltzmann simulations that were in good agreement with the measured data.
The measured permeabilities were in excellent agreement with the values predicted by the well-established Katz-Thompson model (KTM) that relates the permeability to the conductivity and a critical pore diameter lc (diameter of the largest sphere that can pass through the structure) of the structure. The problem with the KTM is that it does not allow a prediction of the permeability without knowledge of its conductivity. We introduced an expression that uses the Euler characteristic χo of the sample-spanning conducting pore space and the number of obstacles N to predict the permeabilities of the investigated structure:
k cl2c =
1−χo
N α
. (8.1)
The expression, which can also be understood as an overlapping probability of individual grains, gave very good agreement with the results of the ROMC and ROME structures.
The question of the universal applicability of this expression naturally arose.
To make the expression universally applicable, an effective number ˆN of obstacles that can be determined from the perimeter, area and Euler characteristic of the pore space, was introduced. After replacing N by ˆN in (8.1), the agreement between measured and predicted values for the ROMC and ROME structures was still very good. For the EROME structures an acceptable agreement was found for high porosities, which, how-ever, deteriorated close to the percolation threshold. In case of the EROMC structures the expression did not yield useful values over practically the whole range of porosities.
So, the expression does not generally hold for arbitrary structures.
A seemingly reasonable explanation for the deviation is given by the occurrence of very small obstacles in the EROME and EROMC structures, which do not occur in the ROMC and ROME structures. These tiny obstacles might not have a strong influence on the flow properties of the structures, but have a strong effect on the value of ˆN.
It still remains an obvious objective for the future to find a universal expression that can relate purely geometrical quantifiers to the permeability of an arbitrary porous struc-ture. Possibly, a reconstruction of the considered porous structure by an appropriately chosen Boolean model could provide more universal agreement.
In another set of experiments and simulations the transport properties of small col-loids in porous media with varying porosities were studied on an individual particle level
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instead of looking at averaged quantities like the permeability. Particles that are trans-ported through a porous medium are advected by the surrounding fluid and also undergo diffusion. Since the fluid velocity field in a porous medium can be highly heterogeneous, particles will travel at different velocities across the sample which can, therefore, give rise to a wide first-passage time distribution (FPTD) of the particles, much wider than would be expected by normal diffusion. The presence of stagnant parts, that is, areas in the medium where the flow velocity is practically zero, can have a huge effect on the resulting FPTDs, as particles can be trapped in such regions for very long times.
Our objective was to shed some light on the question how the structure of the porous medium and the FPTDs are related.
We measured the FPTDs by a semi-experimental approach where we first experimen-tally determined the particle velocity field of three porous structures and then used the velocity fields to perform an overdamped Langevin simulation to get a large number of particle trajectories which allowed us to determine FPTDs. As expected, the FPTDs for a high-porosity structure can all be described by the well-established advection-diffusion equation, which holds for macroscopically homogeneous structures. For low-porosity structures, which also contain stagnant areas, the FPTDs show long-time tails that are independent of the flow rate and can be related to the stagnant parts of the structure.
An equation from statistical physics was used to calculate the mean escape time for every stagnant part:
τ = A πD0
ln P
d + 1
. (8.2)
Here, A is the area of the stagnant part, P its perimeter, and d the size of the small exit pore that connects the stagnant part to the flowing part of the structure.
The longest times that are observed in the FPTDs agree very well with the calculated values for the mean escape times of the largest stagnant areas, which means that infor-mation about the presence and extent of stagnant areas can be obtained by looking at the long-time tails in FPTDs.
In addition, we also performed simulations of active particles, i.e., particles that can propel themselves, in a low-porosity structure. The resulting FPTDs were mainly af-fected in two ways. First, the maximum times compared to the passive particles were lowered, which can, in a certain motility regime, be explained by a higher effective diffu-sion coefficient for active particles since they sample a larger region during the same time and can, consequently, escape out of stagnant regions faster. Second, for active particles with high motility the shortest times in the FPTDs increased due to their ability to leave the main paths where the fluid flow is fastest and travel along regions with lower flow velocities.
In the future it would also be interesting to investigate transport of active particles in structures with a totally different morphology which could, e.g., lead to a slowing down of their transport by trapping in stagnant parts of a particular geometry. Also a more in-depth study of the detailed motion of the active particles in a porous medium and their probabilities of presence could yield useful information, since active agents, like bacteria, are often used to facilitate bioremediation in complex environments. According
8. Conclusion and outlook
to the given structure the agents properties, such as their motility, could be adjusted to increase their efficiency.
Another idea would be to add an external force to the system that pulls, like grav-itation, particles in one direction. In case of passive particles one would expect to see a threshold effect, i.e., particles would only be able to make it through the structure, if a certain flow rate or pressure is reached. When particles additionally have some self-propulsion mechanism they could still cross some critical parts of the structure and be, therefore, transported preferably.
There cannot be any doubt that both the study of the relation between the structure of porous media and their permeability as well as the transport behavior of passive and active particles offers will continue to offer a rich field for decades to come. The facets addressed in this work might indeed have raised more new questions than answered long-standing ones.
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