Effect of time stepping strategy on adjoint-based production optimization

The adjoint gradient method is well recognized for its efficiency in large-scale production optimization. When implemented in a sequential quadratic programming (SQP) algorithm, adjoint gradients enable the construction of a quadratic approximation of the objective function and linear approximation of the nonlinear constraints using just one forward and one backward simulation (with multiple right-hand sides). In this work, the focus is on the performance of the adjoint gradient method with respect to the adaptive time step refinement generated in the underlying forward simulations. First, we demonstrate that the mass transfer in reservoir simulation and, as a consequence, the net-present value (NPV) function are more sensitive to the degree of the time step refinement when using production bottom-hole pressure (BHP) controls than when using production rate controls. Effects of this sensitivity on optimization process are studied using six examples of uniform time stepping with different degrees of refinements. By comparing those examples, we show that corresponding optimal solutions for target production BHPs deviate at early stages of the optimization process. It indicates an inconsistency in the evaluation of the adjoint gradients and NPV function for different time step refinements. Next, we investigate the effects of this inconsistency on the results of a constrained production optimization. Two strategies of nonlinear constraints are considered: (i) nonlinear constraints handled in the optimization process and (ii) constraints applied directly in forward simulations with a common control switch procedure. In both strategies, we observe that the progress of the optimization process is greatly influenced by the degree of the time step refinement after control update. In the case of constrained simulations, the presence of control switches combined with large time steps after control update forces adaptive refinement to vary the time step size significantly. As a result, the inconsistency of the adjoint gradients and NPV values provoke an early termination of the SQP algorithm. In the case of constrained optimization, the inconsistencies in gradient evaluations are less significant, and the performance of the optimization process is governed by a satisfaction of nonlinear constraints in SQP algorithm.