  Quantification of uncertainty in production/injection forecasting is an
important aspect of reservoir simulation studies. Conventional approaches
include intrusive Galerkin-based methods (e.g., generalized polynomial chaos
(gPC) and stochastic collocation (SC) methods) and non-intrusive Monte Carlo
(MC) based methods. Nevertheless, the quantification is conducted in
reformulations of the underlying stochastic PDEs with fixed well controls. If
one wants to take various well control plans into account, expensive
computations need to be repeated for each well design independently. In this
project, we take advantages of the equation-free spirit of convolutional neural
network (CNN) to overcome this challenge and thus achieve the flexibility of
efficient uncertainty quantification with various well controls. We are
interested in the development of surrogate models for uncertainty
quantification and propagation in reservoir simulations using a deep
convolutional encoder-decoder network as an analogue to the image-to-image
regression tasks in computer science. First, a U-Net architecture is applied to
replace conventional expensive deterministic PDE solver. Then we adopt the idea
from shape-guided image generation using variational U-Net and design a new
variational U-Net architecture for "control-guided" reservoir simulation.
Backward propagation is learned in the network to extract the hidden physical
quantities and then predict the future production by the learned forward
propagation using the hidden variable with various well controls. Comparisons
in computational efficiency are made between our proposed CNN approach and
conventional MC approach. Significant improvements in computational speed with
reasonable accuracy loss are observed in the numerical tests.
