# Free Lesson: Archie’s Law

**FREE LECTURE ABOUT ARCHIE’S LAWS – WATER SATURATION EQUATION**

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**[Automatic Transcription]** â€¦ One of the methods that’s now widely used is the Archieâ€™s Water Saturation Equation and all the derivative equations that have a Shale Corrections built in them based on Archie’s work. Mr. Archie sitting in his laboratory around 1940 was wondering what he could learn about rocks in the laboratory that had not yet been discovered and the first thing he discovered was that if he took a piece of rock and measured the resistivity of that piece of rock when it was full of water and compared it to the resistivity of the water itself that was the r0 divided by RW that’s at the top of the screen r0 the resistivity of the rock full of water RW the resistivity of the water divide those two numbers out that ratio turns out to be a constant for a particular chunk of rock you can double the RW and it doubles our zero and so on so he found this constant he called it the formation factor and that was new news at the time 1940 142 when he published he also discovered of course that formation factor is related to porosity now he hope it was a very simple relationship initially he thought it was one over porosity squared and that would have fit with a lot of previous science in in the world many things are related to a power law related to an item to the power to turns out that it’s not a one over porosity squared but a over porosity to the power M and quite often a is defaulted to one and M is default at the two but we can measure these values in the lab by first finding the formation factor and then plotting that data versus the porosity the graph on the right does that formation factor on the vertical axis porosity on the horizontal axis now when you plot the data for a number of different rock samples you’ll see that on log log paper you can fit a straight line through that data the slope of the straight line is M and the intercept of the straight line at the bottom right corner is the value of a you can force the line through an a of one and get a particular slope or you can let the regression line flow freely and end up with the slope being slightly different with a slightly different a value now our cheese data ended up giving us an a of about 0.62 for sandstones and an M of 2.15 those are the world average default numbers widely use today this particular data comes from a project I worked on many years ago probably 3,000 4,000 miles away from where mr. Archie worked and yet the slope of my line 2.18 or 2.19 and the m6 awfully close to the world average numbers so we’re quite confident that we can use world average numbers when we don’t have this laboratory data in sands and Shale sands in the early days the there were no porosity logs so this formation factor concept was used to calculate porosity from resistivity logs the R 0 term came from the shallow resistivity log and RW was found or known and we would end up calculating porosity from our resistivity logs but Archie went further he wanted to understand how water saturation and resistivity were connected and he found the resistivity index he abbreviated with a capital I shown on the top of our screen sometimes abbreviated RI in the literature so resistivity index the resistivity of the rock when it was saturated partly with water and partly with gas air in this laboratory experiment divided by R 0 that was the resistivity the rock when it was full of water and again if he wants the high value of the resistivity index on the vertical axis of a log-log graph and plots water saturation on the horizontal axis we find another straight line and for a single core plug now we can get several data points because we can saturate this core plug with varying amounts of water running from 100% wet down to almost no water and turns out again that all this data can be fit reasonably well by a straight line on log log paper the intercept at the bottom right must be at a resistivity index of 1 angle up from there now transforming that into a useful equation we found that saturation was 1 over the resistivity index to a power of 1 over n turned out that n was mostly near to so this turns out to be the square root of 1 over the reasons to the index looks a little complicated and there are certainly ways to make this look much prettier the slope of the line on the graph is the value of N and we do this work in the laboratory on a regular basis to find the values of a and M in the previous slide and the value of n on this slide default value for n is 2 Â