Capacitance-Resistance model (CRM) has been a useful tool for fast production forecasts for decades. The unique combination of simplicity and physics-based nature in this data-driven approach allowed it to stay as an object of scientific interest and get its own place among other types of models capable of giving predictions on flow rates, such as full-scale 3D reservoir models. However, the model simplicity, assumptions, and limitations does not allow wide application of a conventional CRM in complex field cases. A vast majority of studies on CRM are about overcoming its limitations by introducing new coefficients, modifying the analytical form of the equation, etc. Integrating CRM with rapidly developing artificial intelligence (AI) methods seems to be a logical continuation of model evolution.

Recently introduced Physics-Informed Neural Networks (PINN) can preserve CRM's governing equations and coefficients that gives some insights about wells and formations standing out from other popular machine learning and deep learning methods. Moreover, PINN type models give certain flexibility in the choice of architectures – it means that the model architecture can be changed in a way that may assist in solving different problems. Thus, we introduce end-to-end learning of neural networks (NN) while implying some physical constraints. It is intended to overcome one of the major limitations of CRM, which is obtaining predictions for oil and water production rates from total liquid. This way, the additional training of rough approximation fractional flow models that are either not suitable for the case or may require the knowledge of reservoir properties is not needed.

In this work, the well-known concept of Capacitance-Resistance models appears in a new form, which allows performing history matching rather rapidly, achieving robustness and forecasting liquid, oil and water production rates simultaneously. To test this new approach, several datasets (both synthetic and real) were used. The results obtained by PINN are compared to those obtained by a conventional CRM. By conventional we mean the analytical solution, which was modified by our research group to take into consideration common real field cases such as shut-in wells, workover operations, etc. by introducing dynamic characteristic coefficients [1].

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