The computation of permeability is vital for reservoir characterization because it is a key parameter in the reservoir models used for estimating and optimizing hydrocarbon production. Permeability is routinely predicted as a correlation from near-wellbore formation properties measured through wireline logs. Several such correlations, namely Schlumberger-Doll Research (SDR) permeability and Timur-Coates permeability models using nuclear magnetic resonance (NMR) measurements, K-lambda using mineralogy, and other variants, have often been used, with moderate success. In addition to permeability, the determination of the uncertainties, both epistemic (model) and aleatoric (data), are important for interpreting variations in the predictions of the reservoir models. In this paper, we demonstrate a novel dual deep neural network framework encompassing a Bayesian neural network (BNN) and an artificial neural network (ANN) for determining accurate permeability values along with associated uncertainties.

Deep-learning techniques have been shown to be effective for regression problems but quantifying the uncertainty of their predictions and separating them into the epistemic and aleatoric fractions is still considered challenging. This is especially vital for petrophysical answer products because these algorithms need the ability to flag data from new geological formations that the model was not trained on as “out of distribution” and assign them higher uncertainty. Additionally, the model outputs need sensitivity to heteroscedastic aleatoric noise in the feature space arising due to tool and geological origins. Reducing these uncertainties is key to designing intelligent logging tools and applications, such as automated log interpretation.

In this paper, we train a BNN with NMR and mineralogy data to determine permeability with associated epistemic uncertainty, obtained by determining the posterior weight distributions of the network by using variational inference. This provides us the ability to differentiate in- and out-of-distribution predictions, thereby identifying the suitability of the trained models for application in new geological formations. The errors in the prediction of the BNN are fed into a second ANN trained to correlate the predicted uncertainty to the error of the first BNN. Both networks are trained simultaneously and therefore optimized together to estimate permeability and associated uncertainty.

The machine-learning permeability model is trained on a “ground-truth” core database and demonstrates considerable improvement over traditional SDR and Timur-Coates permeability models on wells from the Ivar Aasen Field. We also demonstrate the value of information (VOI) of different logging measurements by replacing the logs with their median values from nearby wells and studying the increase in the mean square errors.

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