Abstract
We propose to solve a black-oil reservoir optimal control problem with the Direct Multiple Shooting Method (MS). MS allows for parallelization of the simulation time and the handling of output constraints. However, it requires continuity constraints on state variables to couple simulation intervals. The black-oil fluid model, considering volatile oil or wet gas, requires a change of primary variables for simulation. This is a consequence of the absence of a fluid phase due to dissolution or vaporization. Therefore, reservoir simulators parametrize the states with an augmented vector and select primary variables accordingly. However, the augmented state vector and the corresponding change of primary variables are not suitable for the application of MS because the optimization problem formulation must change according to the change of variables. Thus, we propose a minimal state-space variable representation that prevents this shortcoming. We show that there is a bijective mapping between the proposed state-space representation and the augmented state-space. The minimal representation is used for optimization and the augmented representation for simulation, thereby keeping the simulator implementation unchanged. Therefore, the proposed solution is not invasive. Finally, the application of the method is exemplified with benchmark cases involving live oil or wet gas. Both examples emphasize the requirement of output constraints which are efficiently dealt with the MS method.