Production optimization with adjoint models under nonlinear control-state path inequality constraints

The general petroleum-production optimization problem falls into the category of optimal control problems with nonlinear control-state path inequality constraints (i.e., constraints that Must be satisfied at every time step), and it is acknowledged that such path constraints involving state variables can be difficult to handle. Currently, one category of methods implicitly incorporates the constraints into the forward and adjoint equations to address this issue. However, these methods either are impractical for the production optimization problem or require complicated modifications to the forward-model equations (the simulator). Therefore, the usual approach is to formulate this problem as a constrained nonlinear-programming (NLP) problem in which the constraints are calculated explicitly after the dynamic system is solved. The most popular of this category of methods for optimal control problems has been the penalty-function method and its variants, which are, however, extremely inefficient. All other constrained NLP algorithms require a gradient for each constraint, which is impractical for an optimal control problem with path constraints because one adjoint must be solved for each constraint at each time step in every iteration. The authors propose an approximate feasible-direction NLP algorithm based on the objective-function gradient and a combined gradient for the active constraints. This approximate feasible direction is then converted into a true feasible direction by projecting it onto the active constraints and solving the constraints during the forward-model evaluation itself. The approach has various advantages. First, only two adjoint evaluations are required in each iteration. Second, the solutions obtained are feasible (within a specified tolerance) because feasibility is maintained by the forward model itself, implying that any solution can be considered a useful solution. Third, large step sizes are possible during the line search, which may lead to significant reductions in the number of forward- and adjoint-model evaluations and large reductions in the magnitude of the objective function. Through two examples, the authors demonstrate that this algorithm provides a practical and efficient strategy for production optimization with nonlinear path constraints.