Estimation of geological attributes from a well log: An application of hidden Markov chains

Geophysical well logs used in petroleum exploration consist of measurements of physical properties ( such as radioactivity, density, and acoustic velocity) that are digitally recorded at a fixed interval ( typically half a foot) along the length of the exploratory well. The measurements are informative of the unobserved rock type alternations along the well, which is critical for the assessment of petroleum reservoirs. The well log data that are analyzed here are from a North Sea petroleum reservoir where two distinct strata have been identified from large scale seismic data. We apply a hidden Markov chain model to infer properties of the rock type alternations, separately for each stratum. The hidden Markov chain uses Dirichlet prior distributions for the Markov transition probabilities between rock types. The well log measurements, conditional on the unobserved rock types, are modeled using Gaussian distributions. Our analysis provides likelihood estimates of the parameters of the Dirichlet prior and the parameters of the measurement model. For fixed values of the parameter estimates we calculate the posterior distributions for the rock type transition probabilities, given the well log measurement data. We then propagate the model parameter uncertainty into the posterior distributions using resampling from the maximum likelihood model. The resulting distributions can be used to characterize the two reservoir strata and possible differences between them. We believe that our approach to modeling and analysis is novel and well suited to the problem. Our approach has elements in common with empirical Bayes methods in that unspecified parameters are estimated using marginal likelihoods. Additionally, we propagate the parameter uncertainty into the final posterior distributions.