Three Statistical Pitfalls of Phi-K Transforms

Abstract

Summary Phi-k transforms are used widely to predict permeability. Some of the difficulties of this exercise are well identified, such as the homogeneity of the population (rock typing), the matching of cores and logs (especially depth matching), and the problem of permeability upscaling. Not so well-known, however, are the pitfalls of a statistical and geostatistical nature that may create significant biases—always in the same direction—an underestimation of permeability. The passage from Phi to k is performed in three steps: (1) in cored wells, an exponential regression equation is established between core porosity and core permeability k; (2) in uncored wells, log porosity is used instead as input to predict permeability; and (3) the same equation is sometimes used again to populate the cells of a dynamic reservoir model in 3D, where input porosity values are obtained by interpolation. The core-scale regression equation generally underestimates permeability by at least a factor of 2. The origin of the bias lies in the reverse transformation from logarithmic to arithmetic scale. To avoid this pitfall, a new permeability estimator is proposed, based on the quantile curves of the Phi-k crossplot. This estimator is data driven and does not assume a priori any particular functional relationship between Phi and k, such as an exponential-regression function. One of the simplest diagnostic tools to check the agreement between log and core porosity is a crossplot of one against the other. In the absence of bias, the points are expected to be distributed along the y = x line. In reality, they either are or they are not, according to which variable is plotted along the x-axis. This apparent paradox is elucidated by bivariate regression theory and related to the difference of investigated volume between core and log data. Direct input of upscaled cell porosity into an exponential core-scale permeability transform amounts to forcing geometric permeability averaging, which may again lead to serious underestimation of the true upscaled permeability when heterogeneity is significant.