  A new deep-learning-based reduced-order modeling (ROM) framework is proposed
for application in subsurface flow simulation. The reduced-order model is based
on an existing embed-to-control (E2C) framework and includes an auto-encoder,
which projects the system to a low-dimensional subspace, and a linear
transition model, which approximates the evolution of the system states in low
dimension. In addition to the loss function for data mismatch considered in the
original E2C framework, we introduce a physics-based loss function that
penalizes predictions that are inconsistent with the governing flow equations.
The loss function is also modified to emphasize accuracy in key well quantities
of interest (e.g., fluid production rates). The E2C ROM is shown to be very
analogous to an existing ROM, POD-TPWL, which has been extensively developed
for subsurface flow simulation. The new ROM is applied to oil-water flow in a
heterogeneous reservoir, with flow driven by nine wells operating under
time-varying control specifications. A total of 300 high-fidelity training
simulations are performed in the offline stage, and the network training
requires 10-12~minutes on a Tesla V100 GPU node. Online (runtime) computations
achieve speedups of $\mathcal{O}$(1000) relative to full-order simulations.
Extensive test case results, with well controls varied over large ranges, are
presented. Accurate ROM predictions are achieved for global saturation and
pressure fields at particular times, and for injection and production well
responses as a function of time. Error is shown to increase when 100 or 200
(rather than 300) training runs are used to construct the E2C ROM.
