Practical Bayesian Inversions for Rock Composition and Petrophysical Endpoints in Multimineral Analysis

Rock composition can be related to conventional well logs through theoretical equations and petrophysical endpoints. Multimineral analysis is a formation evaluation tool that uses inversions to quantify rock composition from well logs. However, because of data errors and the multivariate selection of petrophysical endpoints, solutions from the multimineral analysis are nonunique. Many plausible realizations exhibit comparable data misfits. Therefore, the uncertainties in rock composition and petrophysical endpoints must be quantified but cannot be fulfilled by deterministic solvers. Stochastic Bayesian methods have been applied to assess the uncertainties, but the high run time, tedious parameter tuning, and need for specific prior information hinder their practical use. We implement Markov chain Monte Carlo with ensemble samplers (MCMCES) to assess the uncertainties of rock composition or petrophysical endpoints in the Bayesian framework. The resultant posterior probability density functions (PDFs) quantify the uncertainties. Our method has fewer tuning parameters and is more efficient in convergence than the conventional random walk Markov chain Monte Carlo (MCMC) methods in high-dimensional problems. We present two independent applications of MCMCES in multimineral analysis. We first apply MCMCES to assess the uncertainties in volume fractions with a suite of well logs and petrophysical endpoints. However, defining the petrophysical endpoints can be challenging in complex geological settings because the values of standard endpoints may not be optimal. Next, we use MCMCES to estimate petrophysical endpoints’ posterior PDFs when the endpoints are uncertain. Our methods provide posterior volume-fraction or petrophysical-endpoint realizations for interpreters to evaluate multimineral solutions. We demonstrate our approach with synthetic and field examples. Reproducible results are supplemented with the paper.