Adjoint formulation and constraint handling for gradient-based optimization of compositional reservoir flow

An adjoint formulation for the gradient-based optimization of oil–gas compositional reservoir simulation problems is presented. The method is implemented within an automatic differentiation-based compositional flow simulator (Stanford’s Automatic Differentiation-based General Purpose Research Simulator, AD-GPRS). The development of adjoint procedures for general compositional problems is much more challenging than for oil–water problems due to the increased complexity of the code and the underlying physics. The treatment of nonlinear constraints, an example of which is a maximum gas rate specification in injection or production wells, when the control variables are well bottom-hole pressures, poses a particular challenge. Two approaches for handling these constraints are presented—a formal treatment within the optimizer and a simpler heuristic treatment in the forward model. The relationship between discrete and continuous adjoint formulations is also elucidated. Results for four example cases of D. Kourounis ( ) · L. J. Durlofsky · K. Aziz Department of Energy Resources Engineering, Stanford University, Stanford, CA 94305-2220, USA e-mail: [email protected] L. J. Durlofsky e-mail: [email protected] K. Aziz e-mail: [email protected] J. D. Jansen Department of Geoscience and Engineering, Delft University of Technology, Delft, Netherlands e-mail: [email protected] Present Address: D. Kourounis Institute of Computational Science, Faculty of Informatics, Università della Svizzera italiana, 6904 Lugano, Switzerland increasing complexity are presented. Improvements in the objective function (cumulative oil produced) relative to reference solutions range from 4.2 to 11.6 %. The heuristic treatment of nonlinear constraints is shown to offer a costeffective means for obtaining feasible solutions, which are, in some cases, better than those obtained using the formal constraint handling procedure.