Treatment of Geometric Constraints in Well Placement Optimization

In well placement optimization, the goal is generally to determine the well configuration that maximizes a profitability or productivity measure. An important requirement in these optimizations is the satisfaction of geometric constraints that limit the locations where wells can be placed. In this work, we introduce and test new constraint-handling treatments within the context of particle swarm optimization
(PSO), a stochastic global search method. These treatments enable the satisfaction
of three types of constraints, namely maximum well length, minimum inter-well distance, and minimum well-to-boundary distance, during the PSO run. Our treatment 'repairs' infeasible solutions by projecting them into feasible space. The repair procedure entails the application of alternating projection using sequential quadratic programming (SQP), along with a perturbation step. This perturbation step acts to shift infeasible solutions out of poor local minima found by SQP. 'Post-evaluation' treatments are also required to handle solutions, within the optimization, that remain infeasible after repair. Three types of post-evaluation treatments are considered in this work - the filter method, and static and dynamic penalty treatments.
The performance of our treatments is assessed for optimizations involving the
placement of multiple deviated wells in 3D models. Two cases (one with 18,225 active cells and one with 42,886 active cells), with irregularly shaped boundaries, are considered. In both cases, PSO without the repair procedure has difficulty finding
reasonable feasible solutions. The repair procedure is shown to render feasible the majority of infeasible solutions, thus providing substantial improvement in optimizer
performance. For the first case, we assess the impact of the parameters in the repair procedure and in the (post-evaluation) penalty treatments. The effect of each
parameter is evaluated based on the resulting median objective function value over
five PSO runs. We observe that the use of smaller-magnitude perturbation, and the
replacement of all infeasible PSO solutions by their repaired versions, leads to the
best overall performance. In addition, the penalty treatments are found to outperform the filter method if suitable penalty parameters are assigned. For both models, the use of the constraint-handling treatment greatly improves optimizer performance compared to PSO without the repair procedure. This is observed over a wide range of parameter values, and for all post-evaluation treatments considered.