Optimal Transformations for Multiple Regression: Application to Permeability Estimation from Well Logs

Conventional Imtrltiple regression for permeability estimation from well logs requires a functional relationship to be presumed. Due to the inexact nature of the relationship between petrophysical variables, it is not always possible to identify the underlying functional form between dependent and independent variables in advance. When large variations in metrological properties arc exhibited, parametric regression often fails or leads to unstable and erroneous results, especially for multi variate cases. In this paper we describe a nonparametric approach for estimating optimal transformations of petrophysical data to obtain the maximum correlation between observed variables. The approach does not require a priori assumptions of a functional form and the optimal transformations are derived solely based on the data set. An iterative procedure involving the ul[ernaling conditional expec[a[ion (ACE) forms the basis of our approach. The power of ACE is illustrated using synthetic as well as field examples. The results clearly demonstrate improved permeability estimation by ACE compared to conventional parametric regression methods. Introduction A critical aspect of reservoir description involves estimating References and illustrations at end of paper. permeability in uncored wells based on well logs and other known petrophysical attributes. A common approach is to develop a permeability-porosity relationship by regressing on data from cored wells and then, to predict permeability in uncored wells from well logs. 1’2 Multiple regression is used when large variations in metrological properties exist (e. g. a wide range in grain sizes, high degree of cementation, diagenetic alteration etc.) and a simple permeability-porosity relationship no longer holds good. However, there are several Iim itations to such an approach. Many of these arise from the inexact nature of the relationship between petrophysical variables and u priori assumptions regarding functional forms used to model the data -all leading to biased estimates. When prediction of permeability extremes is a major concern, the high and low values are enhanced through a weighting scheme in the regression. Besides being subjective in nature, such weighting can cause the prediction to become unstable which leads to erroneous results. Most importantly, conventional regression assumes independent variables to be free of error, which is highly optimistic for geologic and petrophysical data, Jensen and Lake? introduced power transformations for optimization of regression-based permeability y-porosity predictions. The underlying theory is that if the joint probability distribution function (j.p.d.f.) of two variables is binorrnal, (he relationship will be linear,3 Several methods exist to estimate the exponents for power transformation. One method, described by Emerson and Stoto4 and adopted by Jensen and Lake,2 is based on symmetrizing the probability distribution function (p.d.f.). Another method is a trial-anderror approach based on a normal probability plot of the data. By power transforming permeability and porosity separately. the authors are able to improve permeability-porosity correlations. However, using a trial-and-error method for selecting exponents for power transformation is time consuming, and symmetrizing the p,d. f. does not necessarily guarantee a binormal distribution of transformed variables. In addition, there are no indications as to whether power transformations will work for multivariate cases.