Lecture 3: the pore scale

We ended with the question, what is a pore? (Notice in passing that our definition of a “porous medium” carefully avoided this question!)

First consider why Mohanty decided that any definition must be arbitrary. His approach to mapping a porous medium onto a network was essentially to send a “virtual balloon” into the medium. The balloon would inflate and deflate as needed in order to precisely fit, and the location of the balloon’s center would be mapped. The result is a “deformation retract”, a skeleton with information about size at each point. But what happens when:

(1) you have a configuration where a “large pore” is slightly pinched in the middle, meaning that the balloon in the center of one side would overlap with another balloon on the other side? or

(2) a “pore” has little pouches on its walls, such that there are essentially nested pores?

The mapping from (continuous) deformation retract to (discrete) pore network requires decisions that do not have mathematically unambiguous choices. Hence the definition of a pore is, in some sense, arbitrary. A similar conclusion was reached more recently by Lindquist.

But while we can’t define a pore, we’ll keep the word because it is conceptually useful.

What does a “pore” look like? Many conceptual models have been proposed, and certainly none is definitive. Fatt’s models had tubes with specified radius and length, but the pore body had zero volume. Incidentally, he also had pore radius inversely proportional to the length, while others, such as Mualem, have done the opposite. We’ll see later that Mualem’s assumption is more in keeping with soils, and in fact implies a fractal nature to the soils, another useful concept we’ll see later.) Later, Joel Koplik (Creeping flow in two-dimensional networks, J. Fluid Mech. 119: 219-247, 1982) developed the “ball and stick” model (spherical body and cylindrical throat), still in use today. In Koplik’s model the majority of the volume is in the body, whereas all of the resistance to flow is in the throats. Skip Scriven championed the throat with converging-diverging geometry, as it allows stable interfaces during capillary-driven processes, and his student Irwin Sutanto used converging-diverging pores with different wettability properties on each “patch” - a pore-scale model with sub-pore-scale features. As with most concepts, specifics of the application will often suggest one conceptual model or degree of abstraction over others.

Back to Mohanty. After the medium was mapped onto a network (a representation of the connective structure), the network was decorated. Decoration involves assigning numerical values to individual components of the network, for example a volume for the junctions and a radius for the constrictions. Notice that mapping and decorating involve completely different concepts: one deals with geometry, the other with topology:

We’ve seen that the pore-scale models cover a wide range of detail and abstraction. At higher degrees of abstraction the network models sometimes seem to merge with the various continuum models - finite difference, finite element, and so on. But a critical difference is that network models handle connections explicitly, while in continuum models connectivity is implied by simple adjacency. This connectivity is where we need to focus now.

Coordination - the number of bonds meeting at a node (or the mean such across the entire object under investigation) - is a fundamental topological metric in the same sense that angle is a fundamental geometrical metric. The coordination number of a medium is the mean number of bonds at each site.

Common lattices:

Coordination is not everything. For example, there is a huge difference between tree-like networks, in which there is precisely one pathway between any two points, and networks with loops. A network with loops may have multiple pathways between any two points. On a square lattice, for example, there are an infinite number of possible pathways between any two points. If we impose the constraint that a given bond can only be traversed once, the number of pathways is still infinite, but only on an infinite lattice. If each bond can only be traversed in one direction, the number of possible pathways again decreases. Water flow is somewhat like these latter two, while diffusion is more like the original, unconstrained case. So you can think of the number of possible pathways as being a useful characteristic of a network.

If a regular, richly-connected lattice is pruned to be close to the percolation threshold - p dropped from 1.0 to p_c - the number of possible pathways between any two points will decrease from infinity to a few and eventually to zero - the case where at least one of the points is inaccessible.

Can a regular lattice with uniform coordination approximate an irregular or chaotic network with a distribution of coordination? The party line is yes - see works by Winterfeld (Percolation and conduction of random two-dimensional composites, J. Phys. C 14:2361-2376, 1981) and Jerauld (Case studies in topological disorder, J. Phys. C 17:1519-1529 and 17:3429-3439, 1984), but note that these papers ignore pore structure. More recent work suggests that the approximation is fair for some properties (e.g. the percolation threshold) and poor for others (e.g., permeability). Conversations with Peter King (BP-Amoco and Imperial College, London) indicate that the only way to get the “right” answer is to have the network precisely mimic the sample, pore for pore. Nonetheless, a lesser model, even if based on a regular lattice, can give approximations and can reproduce the correct behavior if the geometrical and topological properties have the correct mean, statistical distribution, and spatial correlations.

Lattices with low coordination, e.g. near the percolation threshold, have some unexpected properties. Consider the case of porosity, which we think of as the proportion of the volume that is not solid. But in a sparsely connected medium there are several kinds of porosity:

Notice that this classification of porosity is purely a function of topology: no information on radii is needed to classify a given bit of void space.

What is pore structure? There are no definitive answers or metrics, and I know no approach that is without flaws. My view is that structure necessarily involves both geometry and topology, plus some connection between the two. In other words, pore structure stems from the spatial correlation of pore sizes, the spatial structure of pore coordination, and their interaction (or cross-correlation). This view can be applied both at the scale of the “matrix”, and at a pedon scale so as to include wormholes and fractures.

I’ll leave it to you to ponder why our language and curricula are so much richer in geometry than topology!