--> Estimation of Arithmetic Permeability From a Semi-Log Poro/log10Perm Regression
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# Estimation of Arithmetic Permeability From a Semi-Log Poro/log10Perm Regression

## Abstract

Permeability in geologic model cells is usually estimated using a semi-log Poro:Perm transform based on core plug data, because perm typically has a log-normal distribution (i.e. the logarithms of perm form a Gaussian distribution). Porosity:log10Perm regressions are expressed in the form: (1) log10Kg = (Slope * Porosity) + Intercept By definition, this is a geometric mean regression because it goes through the center of a log10Perm cloud, and therefore seeks the average of logarithms. However, when flow simulation is run on these models, we routinely need to multiply these perms by factors between 3 and 7 in order to achieve a history match. Why? Pore networks in rocks are neither completely random nor perfectly correlated, with reality being somewhere between these extremes. Theoretically, a geometric perm average approximates flow through a random network of pores, and an arithmetic perm average approximates flow through a perfectly correlated network of pores. The geometric mean is therefore the theoretical minimum possible horizontal perm, and the arithmetic average is the theoretical maximum possible horizontal perm. Recall that history matching teaches us that the true perm must be several times the geometric average minimum. A piecewise arithmetic transform can be made by dividing the core plugs into porosity bins, and then calculating the arithmetic mean of the permeability values (in mD, not log10 mD). When plotted on a semi-log crossplot, the arithmetic mean of each porosity bin, when plotted in log10 perm units, will be some nearly-uniform distance above the geometric regression line. This “boost” over the geometric mean is rigorously defined as a function of the standard deviation of the poro-perm point cloud residual around the geometric regression. The equation describing the boost is: (2) log10Ka = log10Kg + 1.1513 * STD2 where: Ka = Arithmetic Average Kg = Geometric Average STD = Standard Deviation of the residual with respect to the trend, in Log10 Perm units To apply this technique, first calculate the log10 of perm. Then find the best-fit semi-log regression through the porosity:log10perm point cloud. Next, calculate the difference in log10 perm units between each point and that regression (the residual). Calculate the standard deviation of the log10 residual, then square it and multiply it by 1.1513. Add this amount to the intercept of the geometric average equation (1), which is equivalent to multiplication because you are adding logarithms.