Long short-term memory-driven forecast of CO2 injection in porous media

The ability to simulate the partial differential equations (PDE's) that govern multiphase flow in porous media is essential for different applications such as geologic sequestration of CO2, groundwater flow monitoring, and hydrocarbon recovery from geologic formations. These multiphase flow problems can be simulated by solving the governing PDE's numerically, using various discretization schemes such as finite elements, finite volumes, spectral methods, etc. More recently, the application of Machine Learning (ML) to approximate the solutions to PDE's has been a very active research area. However, most researchers have focused on the performance of their models within the time-space domain in which the models were trained. In this work, we apply ML techniques to approximate PDE solutions and focus on the forecasting problem outside the training domain. To this end, we use two different ML architectures, the feed forward neural network and the long short-term memory (LSTM)-based neural network, to predict the PDE solutions in future times based on the knowledge of the solutions in the past. The results of our methodology are presented on two example PDE's, namely, a form of PDE that models the underground injection of CO2 and its hyperbolic limit, which is a common benchmark case. In both cases, the LSTM architecture shows a huge potential to predict the solution behavior at future times based on prior data.