Eka Suwartadi

Nonlinear output constraints handling for production optimization of oil reservoirs

Adjoint-based gradient computations for oil reservoirs have been increasingly used in closed-loop reservoir management optimizations. Most constraints in the optimizations are for the control input, which may either be bound constraints or equality constraints. This paper addresses output constraints for both state and control variables. We propose to use a (interior) barrier function approach, where the output constraints are added as a barrier term to the objective function. As we assume there always exist feasible initial control inputs, the method maintains the feasibility of the constraints. Three case examples are presented. The results show that the proposed method is able to preserve the computational efficiency of the adjoint methods.

Waterflooding optimization in uncertain geological scenarios

In conventional waterflooding of an oil field, feedback based optimal control technologies may enable higher oil recovery than with a conventional reactive strategy in which producers are closed based on water breakthrough. To compensate for the inherent geological uncertainties in an oil field, robust optimization has been suggested to improve and robustify optimal control strategies. In robust optimization of an oil reservoir, the water injection and production borehole pressures (bhp) are computed such that the predicted net present value (NPV) of an ensemble of permeability field realizations is maximized. In this paper, we both consider an open-loop optimization scenario, with no feedback, and a closed-loop optimization scenario. The closed-loop scenario is implemented in a moving horizon manner and feedback is obtained using an ensemble Kalman filter for estimation of the permeability field from the production data. For open-loop implementations, previous test case studies presented in the literature, show that a traditional robust optimization strategy (RO) gives a higher expected NPV with lower NPV standard deviation than a conventional reactive strategy. We present and study a test case where the opposite happen: The reactive strategy gives a higher expected NPV with a lower NPV standard deviation than the RO strategy. To improve the RO strategy, we propose a modified robust optimization strategy (modified RO) that can shut in uneconomical producer wells. This strategy inherits the features of both the reactive and the RO strategy. Simulations reveal that the modified RO strategy results in operations with larger returns and less risk than the reactive strategy, the RO strategy, and the certainty equivalent strategy. The returns are measured by the expected NPV and the risk is measured by the standard deviation of the NPV. In closed-loop optimization, we investigate and compare the performance of the RO strategy, the reactive strategy, and the certainty equivalent strategy. The certainty equivalent strategy is based on a single realization of the permeability field. It uses the mean of the ensemble as its permeability field. Simulations reveal that the RO strategy and the certainty equivalent strategy give a higher NPV compared to the reactive strategy. Surprisingly, the RO strategy and the certainty equivalent strategy give similar NPVs. Consequently, the certainty equivalent strategy is preferable in the closed-loop situation as it requires significantly less computational resources than the robust optimization strategy. The similarity of the certainty equivalent and the robust optimization based strategies for the closed-loop situation challenges the intuition of most reservoir engineers. Feedback reduces the uncertainty and this is the reason for the similar performance of the two strategies.

Second-order adjoint-based control for multiphase flow in subsurface oil reservoirs

This paper presents an efficient way to compute second-order gradients by using the adjoint method for PDE-constrained optimization. The gradient thus obtained will then be used in an optimization algorithm. We propose a conjugate gradient combined with the trust-region method, which may have a quadratic convergence rate of Newtons method. Furthermore, we compare the proposed algorithm to a quasi-Newton method (BFGS). We apply the method for production optimization of oil reservoirs. Two numerical cases are presented, showing that our proposed method requires fewer function and gradient evaluations.

A Lagrangian-barrier function for adjoint state constraints optimization of oil reservoirs water flooding

In the secondary phase of oil recovery, water flooding is the most common way to sweep remaining oil in the reservoirs. The process can be regarded as a nonlinear optimization problem. This paper focuses on how to handle state constraints in an adjoint optimization framework for such systems. The state constraints are cast as nonlinear inequality constraints. In the presence of state constraints, adjoint-based gradient optimization methods can loose their efficiency. Moreover, using existing optimization packages one needs to supply Jacobians of the inequality constraints. We propose a Lagrangian-barrier function based method which adds the state constraints as a term to the objective function. Furthermore, we present a numerical case demonstrating that the feasibility and efficiency of the proposed method.

Nonlinear Output Constraints Handling for Production Optimization of Oil Reservoirs

This paper presents a gradient-based optimization method to handle nonlinear output constraints problems for large-scale systems in production optimization of oil reservoirs. The method is based on a barrier function approach, where the output constraints are added as a barrier term to the objective function. The gradient is obtained by using the adjoint method. Further, two case examples are discussed. The results show that the proposed optimization method is able to preserve the efficiency of adjoint methods.