Lecture 2: Approaching the Pore Scale

If a porous medium is a composite of a fluid and a solid, does a single grain of sand (ignoring internal porosity) constitute a porous medium? If so, where are the pores? and if not, what is missing from the definition?

Issues of scale show up in many natural systems, and soils are no exception. We saw earlier that so-called “bulk properties” can’t reasonably be made at a point: they require a non-zero volume. But I also asserted that the statistical distribution of parameter values, and their spatial variability, are important. So there’s a conundrum: measure at too small a length scale, and your measurements will reflect variability but may be meaningless. Measure at too large a scale, and you average out all the variability. To illustrate, let’s consider the scale-of-measurement effect on measured porosity of a bucket of marbles. If I start with a point at the center of a marble and grow my length scale, I’ll see a porosity of zero until my length scale is greater than the marble’s radius. Further increases in length scale will raise the measured porosity, soon going above its bulk value while the high porosity region just beyond the marble’s edge is over-represented. But the fluctuations in measured porosity will decrease as the length scale increases, and eventually I’ll have a pretty stable value. The volume at which the measurement becomes stable is called the Representative Elementary Volume (REV), and it generally has a volume of at least 30 marbles (look at the T-tables in the back of your statistics book, and you’ll see that generally list values for up to N=30; for N>30 the normal curve works fine.).

Notice that the REV for one property may not be the same as the REV for another. And of course, identifying an REV with respect to one property is no guarantee that the property won’t show greater variability at some larger scale.

Returning to the question of whether a single grain of sand constitutes a porous medium: by analogy with the REV concept, let’s require that we have an REV before we can speak of it as a medium. So, for example, 30 grains of sand could be a porous medium, but fewer would not be. Yes, this is arbitrary, but it helps us keep some perspective.

Now we can ask The Big Question: why would one want to look at a porous medium as a bunch of things below the REV, when it’s much easier / more convenient / more intuitive to think of it as one thing above the REV? What can we learn by studying one or ten grains of sand that will help us to understand a whole bucket of sand? Of course we can learn about microscopic stuff – surface properties of sand, mineralogy and roughness and the like – but how will the “bunch of pores” approach help us to arrive at bulk properties, like the permeability of our soil sample?

The answer to this question requires some philosophical and historical perspective. Science is traditionally reductionist: each problem is broken down into component parts, which themselves are then subdivided. Zooming in deeper and deeper, more and more narrowly, is an immensely powerful approach that has served science well for centuries (and resulted in overly specialized PhDs!). On the other hand, there is a new scientific appreciation that reductionism alone is not sufficient: a sort of “systems approach” is needed. The whole is greater than the sum of the parts (sorry, Aristotle). A building is not identical to a pile of bricks, traffic is not simply an isolated driver multiplied by 1000, and a mouse in a blender is different before and after the blender is turned on. The difference between the whole and the sum of the parts is the interaction between them, how the parts relate to each other and thereby produce something new. So we go to the pore scale both to understand what’s happening there (reductionism), and in the hope that understanding the interaction of pores will help us understand what’s happening at the REV scale - that understanding the parts will help us understand the whole (systems approach).

The scientific term for the concept that we’re playing with here is emergence: the idea that the macroscopic behavior or bulk property that we observe is often not predictable from an understanding, even an exhaustive understanding, of a single unit or component. In many cases, like a building or traffic or an organism or the economy, the bulk behavior is driven by the interactions of many smaller-scale units. The bulk property is not obvious - indeed, it may not even exist - at the unit level, but it emerges from the interaction of many units. From a modeling perspective, the emergent property is the final output. For example, given a pore size distribution it might be the permeability.

With that in mind, it makes sense to (at least sometimes) think of soil as a lot of pores, rather than just a box with properties that seem to come from nowhere.

Suppose we think of a porous medium as a (finite) volume, each point of which is, by virtue of being either solid or fluid, potentially important to the behavior of the whole. This concept clearly leads to an infinite regress, implicitly requiring detail at the molecular scale and below. Practicality requires some degree of upscaling from the quarks and gluons, some generalization or abstraction, so let’s move from non-dimensional to the micron scale. Suppose we have used micro-tomography to analyze a rock core at the 1-micron scale, which we then transfer into a simulation model composed of 1-micron cubes each of which is solid or fluid. We could model processes in this medium by having “molecules” zooming around between neighboring fluid cubes, colliding with each other and with the solid phase and so transferring momentum, mass, and energy. This kind of model is called a Lattice-Boltzmann model (LBM), and has tremendous power to model, test, and explain processes at very small scales - though at some cost in terms of computer memory and time (see http://www.agron.iastate.edu/soilphysics/a677_LBM.pdf for more information on LBMs).

A more common level of upscaling or abstraction characterizes the porous medium discretely, pore by pore, where each pore has whatever information is needed: radius, volume, shape, connections to neighbors, and so on. This has the advantage of stopping the infinite regress, at least supposing that we can define what a pore is (or how small a pore can be)! For an excellent introduction to porous media with an emphasis on the pore scale, see FAL Dullien’s Porous media : fluid transport and pore structure (1992, Academic Press). For a classic work on continuum views of porous media, try Jacob Bear’s Dynamics of fluids in porous media (1972, but now available as a Dover publication, 1988). Most soil physics textbooks will focus on the continuum view (almost) exclusively.

I want to emphasize here that continuum and discrete models of porous media are both correct, at least to the extent that any model is “correct”. Every model is wrong in that it falls short of the real thing, but a useful model makes good predictions and/or provides insight into mechanisms. The continuum and discrete perspectives represent different levels of abstraction, and each is more convenient or better suited to some tasks, and less so to others. Likewise, would you ask which is correct: Cartesian or radial coordinates? They are both valid approaches, and generally a given application or need guides the selection of the appropriate model.

The idea of a pore network model was introduced by Irwin Fatt, a petroleum engineering graduate student. In his dissertation, Fatt asserted that existing conceptual models of porous media were inadequate. The bundle-of-capillary-tubes model was mathematically tractable but overly simple, had little explanatory power, and could not reproduce many phenomena such as hysteresis or soil structure. The alternative, a container of glass beads, was still overly simple, and yet it was too complex to be mathematically tractable. Fatt proposed a new model, a pore-network model, and did some early paper-and-pencil simulations of drainage and flow using various regular 2-D pore networks. See Fatt’s three article series on “The network model of porous media”, Petr. Trans. AIME 207:144-177, 1956. Notice that in a pore network, it is precisely the interaction between the pores that results in bulk (emergent) properties, and indeed, Fatt’s model yielded “realistic” saturation / capillary pressure curves, unsaturated conductivity curves, and hysteresis.

Around the time that Fatt developed his network model ideas, a couple of British mathematicians published a paper inspired by work they’d done on gas masks during WWII. The gas masks of the time used granules of activated charcoal, and Broadbent and Hammersley realized that proper function of the mask required careful navigating between two extremes. At one extreme, the charcoal was highly permeable, air flowed easily through the cannister, and the wearer of the mask breathed insufficiently filtered air. At the other extreme, the charcoal pack was nearly impermeable, and while no poisonous gases got through, neither did sufficient air! The optimum was to have high charcoal surface area and tortuous paths for air flow, ensuring sufficient time and contact to absorb the toxin. They realized that this condition would be met if the pathways through the charcoal were sparsely interconnected, and named the mathematical framework that they developed “percolation theory”, because the meandering paths reminded them of water trickling through a coffee percolator.

We’ll look at percolation theory more in a couple of lectures, but here I’ll just give an intuitive introduction. Suppose we have a pond in which lily pads occur at intersections of a square grid, and a curious mouse can jump no farther than the distance from one lily pad to its nearest neighbors. If 100% of the grid intersections have a lily pad, then clearly the mouse can, from any starting point, visit every lily pad in the pond. If we remove at random 33% of the lily pads, the mouse can still cross the pond, and can visit most of the remaining lily pads, but any path longer than a few jumps has increased tortuosity. Now go down to 50% of the lily pads, and you’ll find that the mouse can’t cross the pond at all. Percolation theory states that at some critical probability (approximately 0.59 for our lily pads), called the percolation threshold and written p_c, there will be precisely one pond-crossing pathway. This pathway occupies a smaller proportion of the lily pads in bigger ponds, going to zero for an infinite pond. If we plot the proportion of visitable (accessible) lily pads as a function of pad occurrence probability, we get something like this:

As the percolation threshold p_c is approached from either direction, things change fast: similar sharp transitions are seen in conductivity, diffusivity, tortuosity, and other bulk properties. We will visit this in more detail in lecture 4, but you can already see how it might relate to pore networks and porous media:

The use of pore networks has generated an extensive literature, and the links between abstract-seeming networks and porous media are no longer fundamentally in question. Looking to firm up that link, KK Mohanty showed that, in concept, it is possible to translate from a complete 2D or 3D knowledge of a porous medium to a network of pores and throats. However, he also showed that there is inevitably some element of arbitrariness to how we make this translation - a user-supplied, subjective decision is needed (see Mohanty, Davis, and Scriven, 1980, Physics of oil entrapment in water-wet rock, paper SPE 9406). Philosophically, Mohanty thus leaves us in a difficult position: if we can’t define a pore, how can we even study it? What is a pore, anyway?