- [Instructor] So can we say anything, can we make any predictions about pore geometry porosity correlations? So what we're talking about is actually the derivative of the pore geometry with respect to the porosity. It's not equal to zero obviously. That means that if we change the porosity, the conductivity is going to change. It's also greater than zero and that's obvious as well. It means that if across the increases in the conductivity never decreases it always increases. If it's greater than zero what is it? We don't really know but the simplest assumption we can make is that it's a constant and let's investigate the consequences of assuming it's a constant. The equation we wrote down is the simplest differential equation you can possible write down so it has a simple solution and the solution is that the pore geometrical factor is linearly correlated to the porosity by this equation a not phi plus a not where these constants a not and b not would be determined by boundary conditions but we could say a priority that we expect to see a line in porosity geometrical factor space and the line is gonna have roughly a slope of one and why do we say that? Well this point in the red circle is called the brine point and at 100% porosity the geometrical factor or the efficiency of this conductor is gonna be 100 percent so that point must lie on the line. On the other hand, the conductivity will vanish some porosity still left in the rot so that line must go through the lower left corner or close to it so we can make that prediction graphically. What could we do analytically? Well I'm about to show a slide full of equations but I'm gonna introduce them one at a time so your eyes don't glaze over looking at them. I'm gonna predict the porosity geometry correlation. So here's our assumed model, what we'll call a geometrical factor theory model and if I make the substitution for a geometrical factor then the equation becomes this equation and I want to normalize this equation by the brine conductivity so here we are normalized by the brine conductivity. This is the form that I want to use to apply the boundary conditions, applying the right boundary condition, rbc, you see that the when the ratio sigma not sigma w is equal to one, then one is equal to a not plus b not so there's a constraint on our adjustable parameters. On the left boundary condition then the normalized conductivity is going to be zero and the porosity is not going to be zero it's going to be some value which I call phi sub theta and phi sub theta could be zero and you see that on the righthand side the left factor and if we took that zero we'd have Archie's law that would say that the porosity and the conductivity vanish together. That's not the solution I want to look at. I want to look at the solution where the conductivity is zero but the porosity is not zero so this is the boundary condition that we want to solve and here we're solving for this porosity, phi sub theta at which conductivity vanishes and there's still porosity remaining in the rot. What could we say about this? We know it's going to be a small number so knowing that's going to be a small number, then we could predict from the right boundary condition that a not Is gonna be equal to about one and therefore that the geometrical factor is approximately equal to the porosity so that's our prediction for the correlation between pore geometry and porosity. So let's validate our predicted pore geometry porosity correlation, we're gonna validate it using Archie's Nacatoch Sandstone Data, this is the seminal dataset for the industry. This was published in 1942 and Archie's law was based on this dataset and another one and we see that if we plot E not from the Archie data versus porosity that the trend that we get as coefficients a not at point 92 and b not at point 02 so point 92 is approximately equal to one and point 02 is approximately equal to zero. If you would notice the outliers to the northwest of the main trend, I put a red box around them. These three points have always disturbed me in this dataset. I don't know why they're up there, maybe there's cracks in the rocks or something, but if I assume that those are not really Archie rocks and I take them out of the correlation, then the correlation becomes a not equal 1.03 and b not equal 0.01 so our expectations are even better met in that case but in any case the expectations that we predict in a priory are well met.