Damped natural vibrations of fluid in a well interfaced with a reservoir

We study the natural vibrations of the liquid column in a vertical oil well that occur during a sharp shutdown or opening of the well (water hammer). In this case, the period of vibrations and the intensity of their damping are determined not only by the length and diameter of the liquid column in the well and the rheological properties of the liquid, but also by the reservoir characteristics of the bottomhole formation zone (in particular, permeability coefficients, quality of well perforation and properties of the fractures formed). Solutions to the problem of natural damping vibrations of the liquid column in the well are found using a mathematical model describing the movement of the liquid column in the well and the filtration in the bottomhole zone. Characteristic equations for determining complex frequencies (frequency of vibration and damping coefficient) are obtained. The dependences of vibration frequency, damping coefficient and damping decrement on reservoir permeability values are studied; the amplitude of vibrations at different points of the well is found.