# Percolation theory: Is there a clear relationship between “probability of connection” and “effective porosity”?

Disclaimer: I am a geophysicist, not a physicist. Sometimes we speak a slightly different language and use different terms.

Suppose you have an $N$ x $N$, 2-D rectangular lattice with bonds which have a probability $p$ of being "open" and a probability $(1-p)$ of being "closed". At some threshold probability, $p_c$, there is a 100% probability that there will be a path through the lattice from one side to the other. For the square lattice, $p_c$ = 0.5.

In terms of rocks, the porosity, $\phi$, of a rock is defined as the void (or pore) space (i.e. the "open" space) divided by the total volume (i.e. the "closed" space + "open" space). In the case of the above 2-D lattice, the porosity of a given random mesh will be approximately equal to $p$ itself:

$$\phi \approx p$$

However, the majority of rock physics problems are primarily concerned with effective porosity, $\phi_{eff}$. This is the connected void space divided by the total volume.

My question: Is there a relationship between $p$ and $\phi_{eff}$?

Example:

Below is a figure of a 5x5 network in which the probability of connection is $p$=0.5 and the grid was generated randomly in MATLAB.

The black edges denote "closed" and the red and green edges denote "open". You can count them and find that there are 28 "closed" and 32 "open" for a total of 60 edges on the mesh. So approximately 50% of edges are "open" as expected from probability. The total porosity of this random mesh is $\phi$ = 0.53 because there are 32 open edges (a.k.a pore spaces) out of a total of 60 edges (a.k.a. total volume).

However, you'll notice that, of the open pores, there are 5 green ones which are disconnected from the main network. In geology, these would be known as "isolated" pores. As a result, the effective porosity of the rock would be the number of red edges divided by total space. In other words: $\phi_{eff}$ = (32 - 5)/60 = 0.45.

If I generate 1000 of these types of meshes, the average $\phi$ will approach $p$. Will the average of $\phi_{eff}$ also converge to some value as a function of $p$?

Any help is appreciated.

Cheers

There is no simple analytical relationship between $p$ and $\phi_{eff}$, but for an infinite lattice we expect \begin{align} \phi_{eff}=& \,0 &\textrm{for} \; \phi < \phi_c\\ =&\, \alpha\left(\phi-\phi_c\right)^\beta &\textrm{for} \; \phi\gtrsim \phi_c\\ =&\, \phi &\textrm{for} \; \phi \rightarrow 1 \end{align} where $\alpha$ and $\beta$ are positive constants and $\phi_c$ is the critical porosity (or probability) corresponding to the percolation threshold. (See, for example, "On the relationship between effective and total pore space in sea ice" by Petrich and Langhorne.) These relationships make sense since for small enough $p$ we expect i.e. $\phi_{eff} \rightarrow 0$ because any open edges are likely to be isolated so there is zero percolation, while for high enough probability, $\phi_{eff} \rightarrow \phi$, since the change of an edge being disconnected is tiny. e.g. $\left(1-p\right)^6$ for an infinite square grid, since each edge has 3 connections at each end.
For a finite lattice, as in your example, $\phi_{eff} > 0$ for any $p>0$, although it may be tiny. For example, Liu, Zhang, and Seaton calculated the fractions of accessible and occupied bonds in three-dimensional cubic lattices of dimension $L$. The figure below (based on their Figure 4) shows the effective porosity $\phi_{eff}$ ("accessible bond fraction" $X_A$ in Liu et al.'s notation) vs porosity $\phi$ ("occupied bond fraction" $X$). The observed bond percolation threshold is consistent with the value of 0.24881 expected for a 3-dimensional simple cubic lattice. The difference between effective and total porosity is negligible for porosity > 0.5. The nominal $L=\infty$ data is actually from $L=60$ simulations with pores on the surface excluded, so it differs slightly from the expected threshold behaviour (the dotted line showing $\phi_{eff}\sim\left(\phi-\phi_c\right)^\beta$ for critical percolation exponent $\beta = 0.4188$).