Two main limitations occur when selecting a model with the best predictive accuracy over a set of candidates to extrapolate future production of wells. First, from a Bayesian standpoint, all models are interpretations of the data and thus, they are always incomplete. Second, we could select a suboptimal model with a better bias-variance tradeoff (model selection induced bias). The goal of this work is to combine different models’ predictions to avoid the aforementioned issues. We illustrate the application of the procedure to a set of tight-oil wells of West Texas.

This work aims to combine different models’ predictions using Bayesian statistics. First, we perform Bayesian inference and generate probabilistic production forecasts for four rate-time models (Arps’ hyperbolic, logistic growth, single-phase, and two-phase scaled solution of the diffusivity equation). Second, we evaluate the predictive performance of each model with production data computing the expected log predictive density (elpd) accuracy metric using leave-future-out cross-validation (LFO-CV). Third, we use the Bayesian Bootstrap to make realizations and quantify the uncertainty associated with the estimation of the elpd for each model. We compute a model's weight for the associated expectation of the elpd for each Bayesian Bootstrap realization. Finally, we calculate an average model weight including all the Bayesian Bootstrap realizations.

This study shows that the present model averaging technique produces a distribution of production forecasts with reduced variance when compared to the ones derived from a single rate-time model. The procedure is suitable to integrate model uncertainty in a simple and computationally inexpensive way since it only requires using a sampling technique. The present approach is the Bayesian analog to Bagging (Bootstrap aggregating), but with the following difference: instead of assuming uncertainty in the data collection, here we are accounting for the uncertainty in model predictions. The model ensemble assigns weights to each model according to its predictive performance.

This work illustrates a straightforward way to integrate the uncertainty of models’ predictions using a fully Bayesian approach combining Bayesian leave-future-out cross-validation to evaluate the predictive accuracy of rate-time models, and the Bayesian Bootstrap to account for the uncertainty in those estimates. We present the application to production data of tight-oil wells building an average predictive distribution for the estimated ultimate recovery (EUR).

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