Post mortem: How does the structure of porous media determine the permeability

In my previous blog post, I explain why I would find it useful if researches would shed some light on the history and the circumstances behind their publications. So, this article is about my paper

Christian Scholz, Frank Wirner, Jan Götz, Ulrich Rüde, Gerd E. Schröder-Turk, Klaus Mecke, and Clemens Bechinger
Permeability of porous materials determined from the Euler characteristic, Phys. Rev. Lett. 109, 264504 (2012)

and it’s the closest thing I’ve ever experienced to what enlightenment, if such a thing exists, must feel like.

In life we are always surrounded by porous media, so that it is usually not very difficult to convince anyone of the importance of knowing how such materials behave on a physical level. The properties of ground soil, a classic porous medium, are obviously important to anyone who has ever performed some form of gardening or construction. When you want to brew a coffee it can be quite a challenge to get the grain size, compactification, filter size and flow rate right so that you get the best out of whatever kind of coffee beans you prefer (or you are like me, don’t care too much about the taste as long as it has enough caffeine to keep you going a bit longer).

Now you might ask, what did physicists figure out about porous media that helps us to deal with these questions on an engineering level? One of the many answered to that is a property of fluid flow through porous materials. Quite some time ago a physicist called Henry Darcy realized that when you put twice the pressure on a core of porous material you also double the flow rate, so these two aspects are related linearly. However he also found that the absolute value of the flow rate at a given pressure was quite different, depending on the type of porous material. Today this might seem obvious to us. Getting water through a clogged pipe is obviously more difficult than when it is fresh and clean. Why is that so? Well, again obviously there is an obstacle in the way, so the water needs to flow through the empty spaces within or around it. Clearly this is somehow a property of the empty spaces in a porous medium. But what exactly is it, that determines the ability to let water flow through? Is it the empty volume? Or the surface area? Or could there be something different that we do not intuitively think of?

The first aspect that you need to think about is, what is the minimum requirement for a porous core to let fluid flow through it? Clearly that is, that there must be a path for the fluid to flow through the core. Imagine a piece of polystyrene foam. Polystyrene is a very light-weight porous medium with more than 95% air content. But you will have a hard time pushing water through it in many cases. The reason is, that, despite the high porosity (the relative volume of air) there is typically no nice path for the fluid to flow in such a material. If a material has a well-connected path, which connects two sides, it is called percolating (derived from the coffee percolator). Physicists and mathematicians in the 20th century realized that for porous media that span a certain range of porosities there is typically a threshold that needs to be crossed such that it becomes permeable. Also, and this is the most important aspect, they found that if you know this threshold and also the actual porosity of a material, you can calculate the permeability from a simple equation that has only one free parameter and one parameter that is based entirely on mathematical symmetries, therefore also called universal (because it is the same for all imaginable objects with such symmetries). This is a great result, but there is a crux with the matter. Where exactly is the threshold for a certain porous medium. As it turns out it is not universal at all. The mentioned example, polystyrene foam does is not permeable despite being more than 95% air, but a bucket full of soil with a porosity in the range of 30% to 40% will let water seep away quite easily.

So, close only counts in horseshoes you might think. But in this publication we showed one way on how to get around this by using topological and geometrical properties of an individual porous sample to calculate the permeability. The trick is to find a quantity that behaves similar in dependence on porosity as the permeability, but that is an entirely static topological property of the structure: the Euler characteristic.

The Euler characteristic is a property of geometrical structures that is related to their topology. Imagine a porous structure made of rubber, so that you can easily bend, compress, stretch and twist it. Is there a property of this material that remains the same no matter which of those things you do to the structure? – Yes. It’s the amount of holes and bridges in between hollow spaces inside the material. They might deform, but their number stays the same. In other words, let’s assume you are a tiny molecule in one of these hollow spaces and you can explore all of the volume of this space. No matter how someone might bend or twist the material, you will never be cut of from any part of this pore space (here we assume that you cannot deform the structure so hard that you effectively join or ‘glue’ pore-walls together).

In two dimensions, the Euler characteristic corresponds to the number of hollow parts and solid parts in the material. So when you look at your structure, all we have to count is the number of connected solid and connected hollow (fluid) parts and calculate the difference. How can this be useful to determine the permeability?

Imagine a very hollow porous medium. Just an empty space with a few obstacles obstructing the flow. In such a case, the Euler characteristic is almost equivalent to the number of individual obstacles, for which we assume they are all similar grains of some solid material. When we now increase the number of these obstacles, we know, at some point some of these grains have to join together so we can fit more of them into the pore space. In this case, the number of connected obstacles will become less than the grains that we have put into the material. It is save to assume that at some point all grains will be connected together and we only find very few connected solid parts and hopefully still one connected hollow space. So it is natural to assume that for structures with many densely packed obstacles the difference between connected solid and hollow spaces will not be so different, so the Euler characteristic will be small. For structures with few obstacles the difference between individual obstacles and hollow pore space will be large and the Euler characteristic will be similar to the number of individual grains. Remember the percolation threshold? This was the porosity where we just barely have a connected pore space in the structure. And now we know that close to this threshold the Euler characteristic must be small, while far away it must be large! If we do not know the percolation threshold, maybe we can just use the value of the Euler characteristic instead?

It turns out that for the permeability and certain types of structures this works indeed. Basically the permeability is then proportional to the critical pore cross-section (this gives us the correct physical dimensions) and the Euler characteristic (number of individual obstacles) divided by the number of individual grains.

While in hindsight this seems quite obvious, I remember it was a big a-ha moment for me. I had measurements and simulation results for the permeability of various porous model structures and collected all the geometrical and topological quantities of these structures to see if I could find any meaningful relation. I very vividly remember putting together these numbers, but for most of the relations that I came up with I had some outliers that did not fit my proposed formula. I know today, there were two things that I had to correct for in order to find a relation that the structures were actually obeying. First of all, for the Euler characteristic I had to calculate it only for the accessible or ‘open’ part of the pore space. In particular for structures close to the percolation threshold, many tiny pore spaces that were not accessible from the outside affected the Euler characteristic, but had absolutely no influence on the fluid flow. This ‘repaired’ a lot of nonphysical results that I got initially. Furthermore, the dimension of the problem is given by the cross-section of the ‘critical pore’, which is the smallest pore in the structure that still allows a fully connected pore space. Imagine if you would close up all pores in the structure starting from the smallest pore in the system, at a certain pore size you would eventually fully clog the pore-space so that now fluid can flow through. This is the critical pore space and it’s area defines the dimension of the permeability. When I took these quantities into account, suddenly all my data-points fitted the prediction reasonably. It was a great moment when I made the final plot. I already saw from the values that I calculated that this might work and seeing the final dataset nicely following an equation that I came up with, was very satisfying.

It was a very satisfying moment, not only because of the beauty of the result, but also since the researcher conducting the experiments, i.e. me, had a pretty difficult job. The microfluidic structures used for my experiments had to be manufactured very careful so that they are close enough to the nominal values for which the simulation was conducted (I did not have fancy lithography equipment, everything was optimized ‘by hand’ as good as possible). In particular the critical pore diameter for structures close to the percolation threshold is very sensitive to fabrication uncertainties. Also my samples were very prone to collapse from the capillary forces that arise when injecting fluid into the structure for the first time, due to the small height. It was a bunch of very tedious experiments, but in the end I had a set of 10 samples that properly reflected our numerical models.

The observation of this study came with a pinch of salt though. It should be clear that the formula that I proposed cannot work for arbitrary porous structures. For any kind of model that does not take into account the full microstructure of the porous medium, it is typically easy to find some counter-example structures that do not obey the relation. Typically you can add things like microscopic obstacles and connections somewhere, that clearly do affect the topology, but have almost no influence on the flow. On which length scale do you define pores and open spaces anyway? Do you need to take into account microscopic pores that connect individual pore spaces? Maybe even the individual atoms or the space in between that make up the material? Of course it largely depends on the specific problem. Is it a gas or a fluid that permeates your structure? Is the porous medium saturated or is there gas trapped inside? All this makes a real life porous structure significantly more complicated than the simple models that I studied. However, I still hope that our findings can be incorporated into the multitude of models applied to real porous structures. But even if not, I would say, in this case the famous quote by Richard Feynman applies: ‘Physics is like sex: sure, it may give some practical results, but that’s not why we do it.’

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