A water hammer is an important natural phenomenon that can be used to fracture rock with enhanced local water pressure. The oscillatory injection of a column of water can be used to make a pipe water hammer. However, the optimal injection frequency to create a water hammer has not yet been found. The main reason for this is that the distribution of fluid pressure and its variation are unclear inside a pipe. In this study, we demonstrate for the first time that there can be significant supercharging phenomena and a law governing their appearance in a water-filled pipe. We first find the optimal pulse frequency to reproduce the supercharging process. We also clarify the supercharging mechanism at an optimal frequency. First, a simplified pipe model is adopted, and weakly compressible Navier–Stokes equations are developed to simulate the flow of water in pulse hydraulic fracturing (PHF). The computation code is developed using the MacCormack method, which has second-order accuracy in time and space. The computation codes and program are validated using experimental data of weakly compressible flows. Then, the square pulse effects are studied inside a pipe, including the effects of pulse frequency, amplitude, pipe length, diameter, and wave speed. Finally, a new universal frequency model is built to describe the relationship among optimal pulse frequency, wave speed, and pipe length. The results show that in square PHF, there is a family of frequencies for which the fluid peak pressure can be significantly enhanced, and these frequencies include the optimal pulse frequency. The optimal frequency of a square pulse depends on the pipe length and wave speed. At the optimal pulse frequency, the maximum peak pressure of the fluid can be increased by 100% or more, and cavitation occurs. These new landmark findings are very valuable for understanding pulse supercharging in an internal water wave. In addition, a new universal frequency model is built to predict optimal pulse frequency. This study identifies an evolution law of peak pressure inside a pipe and proposes a practical frequency-control model for the first time, which can provide a theoretical guide for PHF design.

Pulse hydraulic fracturing (PHF) is a key technique for reservoir stimulation. PHF can well accelerate the rupture of rock. However, the supercharging mechanism of PHF is not fully understood. The main reason is that the pressure distribution and its variation, especially the peak pressure characteristics, are unclear inside the pipe and fissure. The present research focuses on the sine pulse applied at the inlet of a pipe or fracture to reveal the variation regularity of peak pressure with the pulse frequency, amplitude, pipe length, diameter and wave speed. First, the weakly compressible Navier–Stokes equations were developed to simulate the variation of fluid pressure. The computation codes were developed using the MacCormack method validated by the existing experimental data. Then, the sine pulse effect was studied inside the pipe and fissure. Last, a new frequency model was built to describe the relationship between the optimal pulse frequency, wave speed and pipe length. The results show that there is a family of frequencies at which the peak pressure of the endpoint can be significantly enhanced and that these frequencies are the optimal pulse frequency. It is found that the optimal pulse frequency depends on the pipe or fissure length and wave speed. At the optimal pulse frequency, the peak pressure at the endpoint can be increased by 100% or more, and the cavitation phenomenon occurs. However, the peak pressure decreases when with the decrease in the pipe diameter and fissure departure due to the friction drag effect of the wall. These new landmark findings are very important for the PHF technique. In addition, a new universal frequency model is built to predict the optimal sine pulse frequency. The present research shows the variation regularity of the fluid pressure inside the pipe and develops a sine frequency-controlled method, providing a potential guide for reservoir stimulation.

Considering the discontinuous square pulse wave and continuous sine pulsating wave, we report a distinctive supercharging phenomenon of fluid in a water-filled semi-enclosed pipe and reveal the supercharging regularity. We demonstrate that there can be significant supercharging phenomena at the pipe end-face if the water is periodically injected at the pipe inlet with certain frequency. Compared to the traditional pulsating injection method, the present injection strategy can remarkably enhance the peak pressure of the water at the end face of the pipe. We explained this phenomenon by numerical simulations based on the computational fluid dynamic method. It’s found that there is a quantitative relationship between the optimal pulse frequency, pipe length and wave speed. The proposed frequency model is suitable for the multi-waveform, such as sine wave, square wave and arcuate wave. The fluid pressure at the pipe end-face intermittently increases and the end-face peak pressure is largest when the inlet injection frequency equals to the optimal frequency.