  We build surrogate models for dynamic 3D subsurface single-phase flow
problems with multiple vertical producing wells. The surrogate model provides
efficient pressure estimation of the entire formation at any timestep given a
stochastic permeability field, arbitrary well locations and penetration
lengths, and a timestep matrix as inputs. The well production rate or bottom
hole pressure can then be determined based on Peaceman's formula. The original
surrogate modeling task is transformed into an image-to-image regression
problem using a convolutional encoder-decoder neural network architecture. The
residual of the governing flow equation in its discretized form is incorporated
into the loss function to impose theoretical guidance on the model training
process. As a result, the accuracy and generalization ability of the trained
surrogate models are significantly improved compared to fully data-driven
models. They are also shown to have flexible extrapolation ability to
permeability fields with different statistics. The surrogate models are used to
conduct uncertainty quantification considering a stochastic permeability field,
as well as to infer unknown permeability information based on limited well
production data and observation data of formation properties. Results are shown
to be in good agreement with traditional numerical simulation tools, but
computational efficiency is dramatically improved.
