Advanced strategies of forward simulation for adjoint-based optimization

Adjoint-based simulation is one of the most efficient methods for reservoir simulation optimization. The gradient information of the objective function and constraints is used to generate a sequence of quadratic programming subproblems converging to the extremum of non-linear problem. The adjoint method provides accurate gradients that help to converge to the optimal solution using the least number of iterations, where each iteration is a forward simulation. The quality and stability of the gradients play important roles in the optimization process. In this paper we present analysis of adjoint-gradients based on different aspects of the forward simulation. We demonstrate that in the presence of compressibility, gradients evaluated using bottom hole pressure (BHP) controls are less consistent with respect to time step refinement, and less stable compared with gradients evaluated using rate controls. Using simple examples, we demonstrate that adjoint-based gradients for rate-controls converge with refinement of the time step while gradients for BHP-controls suffer from convergence problem. Another important aspect of our study is the effect of different nonlinear constraints in the optimization process. In forward simulation, nonlinear constraints often introduce additional complexities due to the discontinuous nature of the switching procedure. Switching can occur at control points in time, or between two controls, and depends strongly on the time-stepping strategy and the truncation error. We compare strategies where individual well constraints are applied directly during the forward simulations and as nonlinear constraints in the optimization process. We demonstrate using two practical examples the advantages and disadvantages of both strategies. We also study the effect of time-truncation error and time-stepping strategy on the quality of the adjoint-gradients. For the time scale, we propose coarsening in both simulation time and redundant control time steps. With larger time steps and smaller numbers of control switches, we can improve efficiency of forward simulation by several fold. Next, the optimal controls of coarse time-step simulation are used as the initial guess for forward simulation of finer time-step resolution. We show how all of these issues affect the optimization of a full-field model.