  The optimization of well locations and controls is an important step in the
design of subsurface flow operations such as oil production or geological CO2
storage. These optimization problems can be computationally expensive, however,
as many potential candidate solutions must be evaluated. In this study, we
propose a graph network surrogate model (GNSM) for optimizing well placement
and controls. The GNSM transforms the flow model into a computational graph
that involves an encoding-processing-decoding architecture. Separate networks
are constructed to provide global predictions for the pressure and saturation
state variables. Model performance is enhanced through the inclusion of the
single-phase steady-state pressure solution as a feature. A multistage
multistep strategy is used for training. The trained GNSM is applied to predict
flow responses in a 2D unstructured model of a channelized reservoir. Results
are presented for a large set of test cases, in which five injection wells and
five production wells are placed randomly throughout the model, with a random
control variable (bottom-hole pressure) assigned to each well. Median relative
error in pressure and saturation for 300 such test cases is 1-2%. The ability
of the trained GNSM to provide accurate predictions for a new (geologically
similar) permeability realization is demonstrated. Finally, the trained GNSM is
used to optimize well locations and controls with a differential evolution
algorithm. GNSM-based optimization results are comparable to those from
simulation-based optimization, with a runtime speedup of a factor of 36. Much
larger speedups are expected if the method is used for robust optimization, in
which each candidate solution is evaluated on multiple geological models.
