Nonlinearly Constrained Well Placement Optimization for Geologic CO2 Storage Using Iterative Latin Hypercube Sampling

Abstract

Summary This work focuses on the well-placement optimization problem, referring to optimizing the well types, well locations, and well controls, for a geological CO2 storage problem with imposed nonlinear constraints to maximize the net present environmental value (NPEV). We present a novel optimization workflow incorporating the Augmented Lagrangian Method (ALM) for solving nonlinearly constrained well placement optimization problems and Iterative Latin Hypercube sampling (TLHS), a gradient-free population-based method. The proposed methodology is referred to as ALM-ILHS. We first benchmark our methodology on the Himmelblau nonlinear toy problem. Then, we consider a mixed-integer NPEV objective function involving integer design variables (well types and well locations) and continuous variables (well controls). The optimization problem solves for optimal values of the well type (injectors or producers), the well locations in terms of (i and j integer coordinates for a 2D problem), and the well controls (injection rates or producing wells’ bottomhole pressures (BHP)) that would maximize NPEV without violating the set nonlinear state constraints for a realistic geologic CO2 storage with brine production. A 2D synthetic saline aquifer model was constructed using a commercial compositional simulator. Besides the bound constraints of the design variables, the nonlinear state constraints such as maximum field CO2 production rate and injection well BHPs are considered. The optimization process involves minimizing the AL objective function using the ILHS. This approach encompasses both inner and outer loop processes with the ALM penalty terms computed for every sample at every optimization iteration. The proposed methodology is also compared with the existing filter method for solving nonlinearly constrained optimization problems. The comparison of the performances of the AL-based method and the filter method on the nonlinear constrained Himmelblau problem show that both methods effectively solve nonlinearly constrained optimization problems. The AL-based method however tends to find better locations more times than the filter method when both methods are independently run for the same number of trials. In addition, the filter method has the advantage of finding several good optimal solution candidates. This provides users the flexibility to select the best value based on their preference for either the objective function or the constraint violation. Results show that opting for all wells as injectors, without any brine producers, gives higher objective function values when there is no constraint imposed on the injection BHP, however, it is more optimal to include producers when the injection BHP needs to be constrained.

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