In the realm of reservoir engineering, we are categorized to be in the small data regime because of the complex physical systems we are dealing with as well as the inherent uncertainty in our problems. The objective of this paper is to present a physics informed neural network (PINN) technique that is able to use information from the fluid flow physics as well as observed data to model the Buckley-Leverett problem.

We use the classical problem of drainage of gas into a water-filled porous medium to test our implementation. The analytical solutions obtained using the method of characteristics are compared with solutions obtained using the DeepXDE library that is built on top of TensorFlow. The automatic-differentiation capability of TensorFlow allows us to interweave the underlying physics of the problem with any observed data to come up with a data-efficient universal function approximator. The flexibility of TensorFlow also allows us to set multiphase parameters to be trainable or not, which would ultimately assist in any inverse modeling efforts using the problem.

Several cases are presented that highlight the importance of the coupling between observed data and physics-informed neural networks for different parameter space. The cases demonstrate the capability of PINNs to augment data-driven solutions. Our results indicate that PINNs are capable of capturing the overall trend of the solution even without observed data but the resolution and accuracy of the solution are improved tremendously once the augmentation of data and physics is implemented. Even with a large mobility ratio, the predicted solution of the PINNs seems promising. The results in our paper indicate that such methods can be utilized to train models that could be used estimate a well-informed initial guess in a reservoir simulator because they capture the overall behavior but miss the intricate details that could be circumvented with conventional reservoir simulation.

The work presented in the paper demonstrates the importance and the capability of applying machine learning approaches in reservoir engineering problems. Additionally, it gives a forward-looking approach for the future of reservoir simulation techniques that could augment data with the physics.

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