So I’ve been discussing percolation and fractals in relation to soil mechanics without giving much background on the topic. I mentioned before that Manfred Schroeder has a chapter about this near the end of his book “Fractals, Chaos, Power Laws.” In that chapter, he uses a simple example of forest fire propagation, which I will elaborate on here.
Consider a 2D uniform grid with points on a 1 m x 1 m spacing. I will use a 250 x 250 grid for this example. Imagine a simple law for forest fire propagation, that if a given tree is on fire, an trees located at immediately adjacent grid points will catch fire in the next time step. If there are no adjacent trees, then the fire will die out. In order for the fire to propagate fully across the grid assuming one side is completely on fire, there must exist at least one fully connected cluster of immediately adjacent trees that spans the complete grid.
If every possible grid location is occupied, then the fire will move across the grid uniformly and reach the other side after 250 time steps.
If none of the initially burning points are connected to other points, the fire dies at the first time step.
Between these two trivial cases, we have a wide range of possibilities.
Consider that a any given point in the grid, the probability of existence of a tree can be defined by some probability function that is the same at all points. If the probability of tree occurrence is 0.5, then we would expect that about half of the grid points will be occupied by trees. The overall density of trees will be roughly the same as the probability of presence of a tree at the individual grid points.
In Schroeder’s book, he suggests that there is a critical density at about 0.59 where percolation will occur, or a complete cluster will exist that spans the complete grid. The following images illustrate this process.
The following image shows the grid with a density of 0.5. Different size clusters are shown with different colours. The whole grid is shown in the main image, and the inset shows a close-up view to illustrate individual clusters. The largest cluster is shown in black in the main image.
The following series of images shows the development of the largest cluster as the density increases from 0.50 past 0.59 to 0.6:
By examining this series of images, one can see right away that we obtain the expected result, namely we have the largest connected cluster span across the complete grid at P=0.59, corresponding to the expected critical density of 0.59.
Theory would suggest that this percolation phenomenon is not scale dependent, and does not depend on the size of the model grid. So even if the grid were 2500 x 2500, or even 25000 x 25000, we should still see the cluster span the complete grid at P=0.59, roughly. You can get some sense of the scale invariance and self-affinity in this final image:
The key idea worth exploiting here in relation to soil mechanics is that near the critical density, the maximum cluster takes on fractal properties, and behaviour becomes markedly different than with either much lower or higher density, where you either have a large number of smaller, disconnected clusters with a reference scale (i.e. mean size), or a single largest cluster with dimensions of the same order as the size of the model. In the “in between” density range, behaviour is dominated by the existence of the fractal character of the largest cluster that just barely spans the model.
In the case of forest fire propagation across the grid, with very simple propagation rules, the time of burning increases exponentially near the critical point due to the fractal shape of the largest cluster. In soil mechanics, we can imagine that the propagation of stresses through the soil matrix (and by extension also through the associated pore fluid medium) will be far more complex at close to the critical density than it would be further away.
More to follow…🙂