We address the problem how to operate the injectors and producers of an oil field so as to maximize the value of the field. Instead of agressively producing and injecting fluids at maximum rate aiming at large short term profits, we are after optimizing the total value (e.g. discounted oil volume) over the whole lifecycle of the field. An essential tool in tackling this optimization problem is the adjoint method from optimal control theory. Starting from a base case reservoir simulation run, this extremely efficient method makes it possible to compute the sensitivities of the total (lifecycle) value with respect to all (time-dependent) well control variables in one go, at a cost less than that of an extra reservoir simulation run. These sensitivities can be used in an optimization loop to iteratively improve well controls.

We implemented the adjoint method and an associated optimization algorithm in our in-house reservoir simulator. In addition to conventional well control options based on the well's pressure or total rate, we have also implemented smart well control options which allow the separate control of individual inflow intervals. Special adaptations of the optimization algorithm were required to allow the inclusion of inequality constraints on well control (pressure and rate constraints).

We applied the optimization algorithm to a number of cases, and found interesting, non-trivial solutions to some optimal waterflood design problems, that would not easily have been found otherwise.

In this paper, we also present a self-contained elementary derivation of the adjoint method, which is different from, but equivalent to the well-known derivation based on the Lagrange formalism.

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