Integrating Markov Chains for Bayesian Estimation of Vertical Facies Sequences through Linear Discriminant Analysis

Abstract

Summary The paper introduces new procedure for vertical facies distribution with respect to well logs and core data in a well from a Sandstone formation by improving Linear Discriminant Analysis (LDA) considering cross-validation and Bayes’ Theorem. The independent variables are gamma rays, formation density, water saturation, shale volume, log porosity, core porosity, and core permeability. LDA has been chosen to estimate the maximum likelihood and minimize the standard error for the nonlinear relationships between facies & core and log data. LDA seeks linear transformation (discriminate function) of both the independent and dependent variables to produce a new set of transformed values that provides a more accurate discrimination with dimensionality reduction. The counts of facies have been formulated through the transition probability matrix of first-order Markov Chains to be prior knowledge into the Bayesian construction. The resulted predicted probability (posterior) has been estimated from LDA based on Baye’s theorem that represents the relationship between posterior with the conditional probability and the prior knowledge. For assessing LDA model, the Cross-validation was considered to check how well the estimation procedure can be expected to perform. The cross-validation results in decrease the squared difference between the estimated and observed facies leading to decrease the uncertainty.