Estimating Water Hammer Damping Ratios Using Continuous Wavelet Transform for Induced Hydraulic Fracture Complexity Characterization

Summary At the end of a hydraulic fracturing treatment, the pump shuts down, causing a water hammer (WH) effect to occur. The WH is influenced by formation properties and fracture complexity. In this study, we use the continuous wavelet transform (CWT) technique to analyze WHs, interpreting damping coefficients of the fracturing stages as indicators of induced fracture complexity within the same reservoir. We introduce a model that treats a WH as a damped harmonic oscillator, leveraging this analogy to understand WH signatures. In this model, the WH signal is viewed as a damped harmonic oscillator. CWT is applied to the post-fracture wellhead signal across multiple scales, using the complex Morlet wavelet. Ridge detection traces dominant frequencies, forming a “ridge” that indicates the system’s natural frequency. Envelope calculation isolates this ridge, capturing amplitude and normalizing it to highlight decay characteristics. The damping coefficient is estimated from the linear regression envelope, where the slope represents the logarithmic decline. In low-permeability reservoirs, fracture complexity and tortuosity strongly influence the damping of the WH signal. The methodology has been applied to various fracture stages, showing a strong correlation between damping coefficients and the number of propagating fractures. This technique automates the estimation of damping coefficients, linking fracture intensity and damping behavior, providing a valuable tool for evaluating induced fracture complexity.