- Welcome. We're met here today to discuss one of my favorite subjects, conductivity in rocks. The title of our talk, A Conductivity Model for Archie Rocks. I'm David Kennedy with PathFinder, a Schlumberger company. My co-author is David Herrick, a colleague that I've known for 20 years. He introduced me to real petrophysics. The motivation for the paper was a couple of letters to the editor that appeared in Petrophysics in 2009. The first was from Robert Freeman, our Schlumberger colleague. Robert was opining that extensive use of empirically derived equations in our industry has its roots in Archie's equation. And the fundamental problem with empirical equations is that reservoir rocks and fluids are too complex and variable to be accurately described by such models. Writing in defense of Archie is one of the grand old men of our profession, Robert Ransom. One of the things that caught my eye and the reason for putting together this research is Robert has said, "At the present time, after 67 years, "many petrophysicists still "do not understand Archie's relationship." And later he says in closing, "At present, simple as it is, Archie's methodology "is far from understood in the industry." This has been my observation as well for the almost 40 years I've been in the industry. And this prompts me to try to explain conductivity in Archie Rocks better than it has been explained. David Herrick and I have written on this extensively. These are eight papers that you can find in the literature, they're all in either Geophysics or Petrophysics or the transactions of the SPWLA. As you Google Herrick and Kennedy you can't fail to find them. A recent paper, Conductivity models for Archie rocks appearing in Geophysics will parallel the talk that I'm about to give. Computers of antiquity are Archie precursors. Back in the day before everyone had a computer on his desk or in his briefcase, all engineers kept a drawer full of graph paper. Typically in my case, it was a stack of linear paper, a stack of log linear paper and a stack of log log paper. The purpose of all of this was to, when data sets came in, to plot them on all of these different kinds of paper, searching for a trend. If I had a straight line on one of these graphs, then the functional form of the data was already determined, and I saved myself a lot of work. These are the references in Archie's paper. So there are four that contain data. The fifth one, the Schlumberger reference is there to reference resistivity logging, but we're interested in the Archie precursors. These are the data sets, graphical data sets from the Archie precursors. There are four of them, Martin, Murray, Gillingham, 1938. Jakosky and Hopper, 1937. Leverett, 1938, and Wyckoff and Botset, 1936. In the upper left corner, we're looking at resistivity plotted against oil saturation. These days we would plot it going the other way. This is resistivity versus water saturation on a log linear scale. In the upper right corner, Jakosky and Hopper also, well, they used a linear scale to plot their resistivity versus water saturation. But they also doubled down and used a log linear scale as well. The two bottom panels, although they look identical, are from different papers. Leverett, 1938, and Wyckoff and Botset, 1936. This is formation conductivity index data versus water saturation. These days these plots would have water saturation on the X axis and would look this way. And the point to be made here is that every kind of graph paper in the drawer was used in the development of Archie's Law, the linear paper, the logarithmic paper. Archie's contribution was to make the plot from his references on log log paper and noticed that they fell upon a straight line. Archie had also defined the formation resistivity factor plotted against porosity on log log paper also fell upon, the trend fell upon a straight line. In log log paper, straight lines are parallels. So, just to summarize, in the development of Archie's Law, every kind of graph paper in the drawer was pulled out and used. Archie's grand synthesis was to notice that the resistivity index from his references could be expressed in terms of a power law on log log graph paper. And that his formation resistivity factor could be expressed in terms of a power law on log log paper. Multiplying the two equations together and canceling terms, in the red box, we find what we now know as Archie's equation or Archie's Law normalized for the brine resistivity. Archie's grand synthesis was a triumph for curve fitting, but I ask the question, where's the physics? Archie himself said, here's the quote, "It should be remembered that the equations given "are not precise, and represent "only approximate relationships. "It is believed, however, "that under favorable conditions their applications "fall within useful limits of accuracy." Well, the second sentence is certainly an understatement. 70 years later, we're still using Archie. However, the first sentence has been ignored. This is my slide, Why petrophysicists cain't git no respect. Now, we all work in the oil industry and we know that the top of every company are either geophysicists or geologists or engineers. But at the top of no company are petrophysicists. Why is that? Well, it's because nobody understands what we're talking about when we present the saturation equation to a room full of our colleagues who are not petrophysicists. And we're always arguing about the meaning of the adjustable parameters in Archie's model. The cementation, exponent m, the saturation, exponent n, and this a, called the tortuosity factor. Now, these days we would write Archie's equation in terms of conductivity where it's a little bit easier to understand. It says the bulk conductivity of an Archie rock in the bottom equation is proportional to the conductivity of the brine in the rock, and the amount of brine in the rock, the amount of brine being VSW, that's the fractional volume of brine in the rock, but this is a complicated function of a fractional brine. I was learning MATLAB the other day and I came across, from this disinterested source, and that's the reason I put it in here, this statement that, you can read it but in red it says, "If your goal fitting the data "is to extract coefficients that have physical meaning, "then it is important that the model reflect "the physics of the data." Physics was not among the goals in Archie's equation. His goal was to fit resistivity and porosity data and predict water saturation. The model was so successful that Archie's successors have assumed it must embody some physics. What is an Archie Rock? So, we won't be talking today about any kind of rocks except Archie rocks as defined here. The main thing about Archie rocks is that there's only one conducting mechanism, and that is conducting brine. So there's no surface conduction due to clay or any other source. The rocks are brine-wet, they're not oil-wet. The grains are relatively equidimensional, the sorting doesn't haven't to perfect, but it's relatively good. The rocks are cemented. And here's some pictures. And so our archetypical example is the Fontainebleau sandstone.