- Now we want to conclude with some connections between the geometrical theory and the Archie's model. Since the pore geometry is not put in by design, where is the pore geometry in the Archie model? Well, the Archie model is this conductivity, and a hydrocarbon-bearing rock is the product of these three factors, brine conductivity, V raised to a power M, and SW raised to a power N. V is a physical, pardon me, SW is a physical property. V is a fractional volume, SW is also a fractional volume. There's no pore geometry in any of those parameters, therefore the pore geometry must reside in the exponents, M and N. This is what Hubert Guyod said in 1944, and this is the origin of our term, cementation exponent. I've underlined the important phrase, Hubert says It is therefore logical to call M the cementation factor, and then he says There's no known method of determining accurately the cementation factor directly. By that, he meant that you have to plot a bunch of points in, observe the slope to get a value for M, the cementation factor. You can't predict what it's going to be without doing that. This is another seminal dataset for the industry, this is Winsaur et al, and the familiar thing out of this is that the Humble Formula is defined in this paper. And, as you have copied the Humble Formula out of the paper and put it in here, it's F equal point oh six two times porosity to the minus two point one five power. The dataset here is published, and it was from nine states, and it was from nine geological epochs that covered three geological eras. And, this was a very popular formula in the 50s when it came out, it was embraced widely, and everyone used it. The only reason that I can think that that happened, is it was the non-Shell formula, and people were eager for something else. Today, this paper would not pass peer-review, wouldn't pass peer-review, because this data set is drawn from nine states, and nine geological epochs, and it doesn't go together. Today, we expect, when we're determining a value for the A, and a value for M that the dataset come from the same formation in order to be meaningful. However, you will still find this formula in the chart book, today. Here is an attempt to derive Archie's Law from a first principle. This is by Pabitra Sen, from Sclumberger-Doll Research, he has a section in his article, the Derivation of Archies' Law from First Principles. He begins with a mixing rule, it's called Bruggeman's Rule, and is Bruggeman's Rule a first principle? Well, that's arguable, but if you can read in the equation, in the middle of the left column there, sigmus of M is the conductivity of the matrix. For an Archie rock, the conductivity of the matrix would be zero, so if you set that factor equal to zero, and do the algebra, then you would find that the normalized conductivity is equal to the three halves power of porosity. So that was a claim, that this was Archie's Law derived from a first principle. Well, it's a matter of taste whether it's Archie's Law derived from a first principle. Is Breggeman's Law a first principle? Is something with three halves instead of two an Archie Law? It's not very rock-like. Now, back in the day, when people were first trying to understand Archie's Law, in terms of physics, they did a lot of physical experiments, and this is taken from the literature. This is showing several datasets, one is for spheres, for rounded quartz sand, for Platy sand, and for other cases. Now, you can see that the porosities, if you look at the porosity scale, these are all very un-rock like porosities. You might wonder where this data fits in the context of a rock. Well, here's Archie's rock. Here's the datasets, so it leaves me wondering Did we learn anything by doing these experiments about conductivity and Archie rock? Why is M equal to two? Well we have, in the geometrical factor theory, an explanation. So, I start with this long equation at the top, which just expands the conductivity in a rock in terms of the three factors. Substitute for the geometrical factor theory, or the geometrical factor, in terms of this linear relationship. Do the multiplication, and notice that A naut is approximately equal to one, and B naut is approximately equal to zero. So that must mean that M is approximately equal to two. In terms of the percolation theory, the formulation of the geometrical factor theory, of course you get the same result. But, I love this because it's just so obvious here. If V sub-theta is almost zero, then you have V times V it's almost V squared. One final comment on M, and M porosity and pore geometry are coupled in a relatively complicated fashion, and we can investigate that. We look at the normalized rock conductivity, in terms of geometrical factor theory and in terms of Archie's model. We can take the log rhythm of those equations and solve for M. So, solving for M, we see that M is the sum of a constant, one, and the ratio of the log rhythm of the geometrical factor to the log rhythm of porosity. So M incorporates pore geometry, but it's not purely a pore geometric factor. This gives insight into why M has, historically, been so difficult to characterize. And since the porosity is almost equal to the geometrical factor, in many cases, it's another illustration of why M is almost equal to two. When data no longer obeys a power law, that is for some low porosity carbonates, to what lengths ought we to strive in order to preserve the power law formulation? Most people will have heard of variable M, which is used in carbonates. If variable M preserves the appearance of a power law, it's actually cheating, because once the exponent is not a constant, we don't have a power law anymore. In fact, we could argue we should just set up an ordered pair of data, and then turbulate between the ordered pairs for porosities that we haven't measured, that's as good a function as any, and it doesn't try to impute any physics to a fitting function. Here we show a formation resistivity factor porosity plot, using the geometrical factor theory to compute curves for various values of the pseudo-percolation threshold. So, we see that some of the curves tend to go up, these are curves where the resistivity would go to infinity with decreasing the porosity. Other curves tend to bend down, these are cases where the, where you would have surface conductivity, and I want to put on here, the shell variable M exponent. So this is the shell equation for variable M, you could find it in the chart book, but if we use the percolation theory model of the geometrical factor theory, we can see that we can fit this curve equally well with a pseudo-percolation factor of minus zero point oh two six. So it's not necessary to introduce variable Ms in order to explain data, if we have a theory. Now, something I've always wondered about, is why was it that the Archie Law was never questioned to this day, as to it's physical significance? So, the Archie Law, the formation resistivity factor is proportional roughly to one over porosity squared. Remember we talked about the fitting coefficient of point nine, something for A and almost point two, or almost two for M? Well, nobody really knew why those numbers were close to one and close to two. And, here's what I'm just speculating, that we had the example of Coulomb's Force Law, which is an inverse square law, and we had the example of Newton's Law, which is an inverse square law, and we had Archie's Law, which was an inverse square law, it must have seemed like there was some cosmic significance to the values of these exponents, and therefore, there must have been some physics involved. Now, the resistivity formulation is an impediment to understanding conductivity in rocks. The bulk resistivity increases in proportion to brine resistivity. But, bulk resistivity decreases in proportion to brine volume. So, Archie was able to normalize, for the effects of brine volume, by dividing the resistivity of the bulk rock by the resistivity of the brine. But, how could he normalize for the porosity? Well, it just isn't clear in that formulation. On the other hand, with the conductivity formulation, the bulk conductivity increases in proportion to brine conductivity, and the bulk conductivity increases in proportion to the brine volume. So, the brine conductivity factor is had by dividing the bulk rock conductivity by the brine conductivity, but then it's obvious how to normalize for the effects of porosity, you simply divide by the porosity. And the geometrical factor then is the bulk conductivity divided, or normalized, by the porosity and the brine conductivity. What Archie could have done, but of course it would never occur to anyone to do this, is multiply through his formation resistivity factor equation by the porosity. Multiplying through by the porosity, you would notice in the red boxes, that in conductivity space you would recognize the same two terms, except one is reciprocated with the other, so here's VF is proportional to the reciprocal of the geometrical factor. And that F, the formation resistivity factor, finally is seeming to be a function of both pore geometry and porosity. So, that is what has made the Archie formulation of the problem very difficult to understand.