Adjoint-Gradient-Based production optimization with the augmented Lagragian method

The production optimization step of the "closed-loop" reservoir management is an optimal well control problem determining optimal operating conditions to maximize hydrocarbon extraction or net present value (NPV) for the remaining expected life of a reservoir. The most challenging part of production optimization is to honor the nonlinear constraints or state-control constraints, such as WOR, GOR and production rates. In this research, we implemented an augmented Lagrangian method for solving the production optimization problem under linear and nonlinear constraints. In our implementation, the objective function to be maximized is defined as the augmented Lagrangian function consisting of the NPV and all constraints except the bound constraints. At each iteration of the optimization procedure, the objective function is approximated by a quadratic model based on the adjoint gradient and the approximate Hessian matrix obtained using a quasi-Newton method. The quadratic model is then maximized subject to the bound constraints using a gradient-projection trust-region method. This step ensures all the bound constraints are satisfied. Once the controls that maximize the quadratic function are obtained at this iteration, we update the Lagrange multipliers or penalty parameter depending on how well the constraints are satisfied, and move to the next iteration. The above process is repeated until convergence. The advantage of the above procedure is that the bound constraints are easily handled using the gradient-projection method for a quadratic approximation of the objective function. Compared to the generalized reduced gradient (GRG) method which is implemented in Eclipse 300, our method does not require the controls to be feasible at every iteration, but the constraints are satisfied within a reasonable tolerance at convergence.;We extend the augmented Lagrangian method to solve the robust production optimization problem. The technique is applied to synthetic reservoir problems to demonstrate its efficiency and robustness. When reservoir description is uncertain, experiments show that the optimal NPV obtained based on a single reservoir model may not be the optimal NPV for the true geology, whereas the application of robust optimization significantly reduces this risk. Another challenging problem for production optimization is to solve multi-objective optimization problems, such as long-term and short-term optimization. Robust long-term optimization maximizes the expected life-cycle net-present value (NPV) over a set of geological models, which represent the uncertainty of reservoir description. As the life-cycle optimal controls may be in conflict with the operator's objective of maximizing short-time production, the method is adapted to maximize the expectation of short-term NPV over the next one or two years subject to the constraint that the life-cycle NPV will not be substantially decreased. Experimental results also show robust sequential optimization on each short-term period is not able to achieve an expected life-cycle NPV as high as the one obtained with robust long-term optimization.