Frequency response of a fixed–fixed pipe immersed in viscous fluids, conveying internal steady flow

Abstract Novel methods are desired to harvest and store power in harsh environments, like those found at the bottom of production wells, to power commercially available monitoring devices. These systems must not only be mechanically robust but also operationally resilient, capable of sufficient power output under the widely varying conditions expected over the service life of a well. Since energy harvesting systems are heavily dependent on natural frequency, this broad range of conditions and/or well configurations makes the design of a suitable energy harvester challenging. Although the American Petroleum Institute (API) has set standards on some of the system variables, other variables are less well defined and may be time dependent. A first step towards the design of an energy harvesting system, then, is to investigate the changes in the natural frequency of a well by varying those inputs possessing moderate to high uncertainty. An analytical model is formed using Euler–Bernoulli beam theory to model the coupled fluid-structural system found in a producing well. A hydrodynamic function is included in the formulation to model the effects of the viscous fluid filled annulus. Due to the form of the hydrodynamic function, the systems natural frequency is solved in the frequency domain using the spectral element method; a method for calculating the displacement response to an external force is also provided. A parametric study is performed to determine the effect various inputs have on a systems first natural frequency. The key inputs considered are the axial force in the production tube, the conveyed fluid velocity, and the hydrodynamic function, itself a function of the annulus fluid viscosity and geometry. The study׳s results are in-line with expectations based on previous publications investigating component wise analogous systems. The inclusion of an axial force shifts the natural frequency of the system and the conveyed fluid velocity at which divergence occurs. The added mass introduced by the real part of the hydrodynamic function causes a shift in natural frequency but not in the bifurcation point. Viscous effects generated by the imaginary part of the hydrodynamic function act to shift the natural frequency of the system and the bifurcation point (This publication approved by sponsor for release, LA-UR-14-27597)..

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