  In this work, an efficient physics-constrained deep learning model is
developed for solving multiphase flow in 3D heterogeneous porous media. The
model fully leverages the spatial topology predictive capability of
convolutional neural networks, and is coupled with an efficient
continuity-based smoother to predict flow responses that need spatial
continuity. Furthermore, the transient regions are penalized to steer the
training process such that the model can accurately capture flow in these
regions. The model takes inputs including properties of porous media, fluid
properties and well controls, and predicts the temporal-spatial evolution of
the state variables (pressure and saturation). While maintaining the continuity
of fluid flow, the 3D spatial domain is decomposed into 2D images for reducing
training cost, and the decomposition results in an increased number of training
data samples and better training efficiency. Additionally, a surrogate model is
separately constructed as a postprocessor to calculate well flow rate based on
the predictions of state variables from the deep learning model. We use the
example of CO2 injection into saline aquifers, and apply the
physics-constrained deep learning model that is trained from physics-based
simulation data and emulates the physics process. The model performs prediction
with a speedup of ~1400 times compared to physics-based simulations, and the
average temporal errors of predicted pressure and saturation plumes are 0.27%
and 0.099% respectively. Furthermore, water production rate is efficiently
predicted by a surrogate model for well flow rate, with a mean error less than
5%. Therefore, with its unique scheme to cope with the fidelity in fluid flow
in porous media, the physics-constrained deep learning model can become an
efficient predictive model for computationally demanding inverse problems or
other coupled processes.
