Optimizing Perforation Phasing through Nearest Neighbor Lookup and Delaunay Triangulation
Authors: Sheikholeslam A. , Jalali S. E. , Ramezanzadeh A. , Shojaei H. 
Published: December 2024

Summary
Sand production is a common occurrence in reservoirs with weak sandstone undergoing pressure depletion during hydrocarbon extraction. Among available techniques, the adoption of the cased, cemented, and perforated well completions (CCPs) stands out for its inherent safety measures and cost-effectiveness compared to alternative methods. This approach is anticipated to offer substantial potential for bolstering hydrocarbon yields when juxtaposed with the openhole completion methodology, positioning it as a preferable well completion strategy for such reservoirs. In the design of perforation arrangements, the determination of shots per linear foot (SPF, spf) is typically contingent upon the targeted hydrocarbon production capacity. Thus, fine-tuning the phase angle becomes imperative to forestall overlap within the damaged zones surrounding neighboring perforations, thereby mitigating the interaction between them—a pivotal consideration in the design optimization process. In light of the absence of constraints and the relative ease of implementing a single helical perforation pattern (SHPP) over other intricate designs, this research is centered on ascertaining the minimal separation between adjacent perforations, termed the minimum perforation-to-perforation spacing (PPSm), across various phase angles within this pattern. This objective is pursued through a swift exploration of nearest neighbors (NN) facilitated by a spatial-partitioning data structure, specifically a k-dimensional tree (KDTree), aimed at probing the optimal phase angles for customary values of SPF, predicated on a well diameter of 8½ in. Achieving a uniform distribution of the perforations has been realized through the application of mathematical and statistical principles, utilizing a parameter known as the equilateral likeness score (ELS). Employing the Delaunay triangulation, the arrangement of the perforations has been meticulously configured to minimize this score, thereby signifying the most even distribution of them. The optimal phase angles have been discerned from two distinct vantage points—maximizing the spacing between adjacent perforations and maximizing the feasible angle values for the phasing. A comparative analysis between the outcomes derived from the proposed theory and those from prior theories—including the isosceles obtuse triangle pattern (IOTP)—reveals the likelihood of notable disparities in the calculated optimal phase angles. Particularly within the realm of the two-wrapped-based methodologies, variations in the magnitude of these angles can be substantial, extending up to approximately 80° for certain shot densities. Notably, the optimal phase angles of 97° and 77° have been estimated for two prevalent shot densities of 6 spf and 12 spf, respectively, contrasting with the values of 99° and 143° previously obtained using IOTP. This substantial difference in phase angles observed for the shot density of 12 spf has been minimized to approximately 1° through refinement of the previous method, achieved by removing its constraint on the number of wraps.