- The geometrical factor theory has an interpretation for suggestible parameters. You consider in this diagram, the rock sample labeled as "a" above, we take the porosity and put it into a tube. Then the porosity is the fractional, cross-sectional area of the plug that is uh void, void space. E naughts of Phi is a number less than Phi. It's the fractional cross sectional area of the porosity that participates in conduction. So what the geometrical factor theory is describing is a partially occluded tube. That's the physical model. Physical interpretation of the adjustable parameters in Archie's model. Well, here's Archie's model. With the equation that has a tortuosity factor and cementation exponent. There they are defined. What do the names mean, what exactly does cementation exponent have to do with the cement? And what exactly does tortuosity factor have to do with tortuosity? Well, no one can really tell you. On the other hand, here's the geometrical factor theory. I'm showing it with, as uh, as normalized conductivity is equal to the sum of E naught Phi plus C naught. C naught is an excess conductivity, which in Archie rocks is equal to zero, but which in rocks that had surface conduction would have some value greater than zero. We saw that uh, the geometrical factor is correlated to porosity, linear with uh, correlation coefficients A naught and B naught. A naught is the rate of change of cross sectional area with a change in porosity. B naught is an irreducible cross sectional area independent of porosity that doesn't participate in conduction. And there's this thing that I call a percolation threshold, which is the value of porosity at which conductivity vanishes. Now I wanna make a connection between percolation theory and the geometrical factory theory of conductivity in rocks. This is the geometrical factor correlated to porosity with uh, adjustable parameters A naught and B naught. And I wanna look at the case of what I call the left boundary condition where the normalized conductivity was zero and the porosity is not zero in this rock. So we had solved for this thing called the percolation threshold. Phi of Theta which was B naught over A naught. Now going back to the geomtrical factor representation and factoring out A naught, we see that the term left inside the, inside the parentheses is B naught over A naught. Well that was just the, what I call the percolation threshold. Now the question is, can I solve for this uh, constant A naught? Well I can solve for the constant A naught if I look at the boundary condition on the uh, on the right so, the normalized conductivity and the geometrical factor theory will be equal to one and the porosity will be equal to one. So A naught satisfies this condition, and solve for A naught. We would write it this way, and we plug A naught back into our equation for the geometrical factor theory. So we see that the geometrical factor has this representation Phi minus Phi of Theta over one minus Phi of Theta. There it is. And therefore our geometrical factor theory can be expressed in terms of normalized conductivity by this equation. Now why did I do this? Well I'll show you in a moment. First, let's look at some of the perceps of um, percolation theory. This is a um, unit cell of a resistor network with resistors radiating from a node. North, south, east, and west, and up and down. And this shows this unit cell put together in a network of resistors. An extensive network of resistors. When all the resistors are present, then the network has maximum conductivity. So now we imagine plucking out nodes one at a time and recording the conductivity as the nodes are plucked out. Each time you pluck out a node, it removes six resistors, and the conductivity is reduced. The right, uh, expression here is from Scott Kirkpatrick's 1973 paper on percolation theory what it shows is for the resistor network, that the normalized conductivity of network is proportional to the probability that the node will be occupied, times this uh, this factor which is a portion, probability minus two over Z. Z is a coordination number. In this case it would be six. So two over six is one third, that number is one third. Compared to the geometrical factor theory, which is isomorphic or formally the same form. Well it really means the same thing. If you think about what porosity, uh, what's probability in a rock. If you give me um, a coordinate X Y and Z, and I put that coordinate in a porous rock, the probability that will fall in a pore is equal to the porosity. So really these are the same formulas. So that's for the resistivity network, and this is for the pore network. These two things in my mind have to be connected. Here's a chart, modified from the literature, and what it's showing is comparisons between the percolation theory and the geometrical factor theory. The uh, red-green line that you see on the lower left is two functions, they plot one on top of the other. One is the geometrical factor theory with a pseudo-percolation threshold of one third, and the other is uh, the Kirkpatrick formula. You can see that, well they were the same formula. They're the same curve. Uh, also shown, as you can see the blue curve is Archie's curve, Archie's Law or M equal two and A equal one, and the red curve just below it, is uh, the geometrical factor theory with a pseudo-percolation threshold of point five. So you can see that they're very close. And uh, we'll not talk about the Bruggeman curves so, but they're mentioned in our papers.