Model selection for error generalization in history matching

In order to better understand reservoir behavior, reservoir engineers make sure that the model fits the data appropriately. The question of how well a model fits the data is described by a match quality function carrying assumptions about data. From a statistical perspective, improper assumptions about the underlying model may lead to misleading belief about the future response of reservoir models. For instance, a simple linear regression model may have a fair fit to available data, yet fail to predict well. On the contrary, a model may perfectly match the data but make poor prediction (i.e. overfitting). In both cases, the regression model mean response will be far from the true response of the reservoir variables and will cause poor decision making. Therefore, a suitable model has to provide balance between the goodness of the fitted model and the model complexity. In the model selection problem, realistic assumptions concerning the details of model specification are the key elements in learning from data. With regard to conventional history match scheme, the data fitting is usually performed by linear least-squares regression model (LSQ) which makes simple, yet often unrealistic, assumptions about the discrepancy between the model output and the measured values. The linear LSQ model ignores any likely correlation structure in discrepancy, changes in mean and pattern similarities reflecting on poor prediction. In this work, we interpret the model selection problem in data-driven settings that enables us to first interpolate the error in history period, and second propagate it towards unseen data (i.e. error generalization). The error models constructed by inferring parameters of selected models can predict the response variable (e.g. oil rate) at any point in input space (e.g. time) with corresponding generalization uncertainty. These models are inferred through training/validation data set and further compared in terms of average generalization error on test set. Our results demonstrate how incorporating different correlation structures of errors improves predictive performance of the model for the deterministic aspect of the reservoir modelling. In addition, our findings based on different inference of selected error models highlight an enormous failure in prediction by improper models.