The rise of modern machine learning has inspired many applications in various fields including petroleum. Many researchers have recently tried utilizing machine learning in general and deep learning in specific for petroleum production time series forecast. One challenge of this task is that production data can exhibit complex non-stationarity in which different patterns emerges in different time frames. In this paper, the ability of a purely data driven deep learning model to handle non-stationary production time series is investigated and a hybrid model that incorporates physics principles into deep learning is proposed to forecast production in tight reservoirs.

In the purely data driven approach, model training is driven solely by a data-based cost function which reflects the differences between model predictions and observations. In the proposed approach, a physics-based cost function is also included in addition to the data-based one. This cost function is derived from the governing differential equations that represent mass and momentum conservation principles of fluid flow in porous media and initial and boundary conditions. The purpose is to guide the model towards learning physically reasonable patterns apart from only fitting observations during training. The physics-based term used is derived from Wattenbarger analytical model and in essence, no analytical solution is needed.

The data used for training and testing are generated by numerical simulations with different geometry parameters and bottom hole pressures. Feedforward and recurrent neural networks are trained on historical production data to predict future responses. The physics-constrained model is compared with the purely data-driven model in several scenarios that have different data quality and boundary conditions. Root mean squared relative error, coefficient of determination, and normalized cumulative production are used as metrics for comparison. The physics-constrained model is observed to be more robust to noise and capable of handling complex non-stationary patterns often encountered in production time series in tight reservoirs due to changes in flow regimes.

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