Optimal modeling of initial expansion followed by contraction nozzle for hydraulic sand-jetting perforation

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Abstract
Hydraulic sand-jetting perforation is widely employed in oil and gas reservoir stimulation and downhole operations, where the nozzle profile design is crucial for enhancing jet energy utilization and improving perforation quality. However, traditional nozzle design often relies on empirical methods and trial-and-error approaches, making it challenging to achieve an optimal contour that balances high efficiency and low energy loss. Against this background, this study focuses on the working conditions of hydraulic sand-jetting perforation and utilizes the Euler–Lagrange (EL) model in combination with the Shooting method to theoretically model and numerically solve for the optimal converging nozzle profile. By formulating a variational optimization problem and incorporating comparisons of turbulent kinetic energy, analyses of pressure and velocity distributions, as well as comparative simulations of different nozzle contours (including a straight conical nozzle, a hyperbolic nozzle, a Vitoshinsky-profiled nozzle, and the EL-based nozzle), the results show that the Vitoshinsky and hyperbolic nozzles achieve increases in outlet velocity of approximately 6.81% and 5.48%, respectively, compared to the straight conical nozzle. In contrast, due to its initial expansion followed by contraction, the EL nozzle experiences a slight reduction in outlet velocity (around 5.82%) while maintaining a more stable internal flow field and a shortened local acceleration zone. This configuration facilitates more rational energy distribution and reduces turbulence-induced losses. The findings provide feasible theoretical and engineering guidance for nozzle profile optimization, laying a foundation for improving the efficiency of hydraulic sand-jetting perforation and advancing nozzle design.

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