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Parameters of unsteady movement of injected fluid in tubing string during operation of pulsing devices
During the operation of pulsing devices on the well face there are short-term overlappings of a through bore for liquid pass as a result of which water hammers appear in the column of injected liquid, the parameters of which depending on the difference of elasticity of liquid and the tubing are transferred to the last one. In this regard there is a need to define the analytical dependences characterizing the values of pressure and speed of movement of liquid in any arbitrary cross section of a liquid column inside the tubing. To study these processes a problem of distribution of a water hammer of viscous liquid in the tubing from operation of pulsing devices on the well face on the basis of N.E. Zhukovsky’s classical equation about a water hammer was solved. Linearized equations of unsteady liquid movement represent a linear hyperbolic system and are a special case of telegraph equations. The problem was solved by the variable separation method which consists in finding particular solutions of the considered system meeting the set conditions. Entry and boundary conditions taking into account the unit at the bottom end of the tubing of reformative pulsing devices are set, and the column is isolated by a packer. The common decision of non-uniform system, according to the known formulas, is decomposed in Fourier series. As a result, we obtained expressions for speed of propagation and pressure amplitudes of hydroblows of injected liquid from operation of pulsing devices in the tubing on the well face. The received expressions were analyzed resulting in regularity of determination of amplitude of hydroblow pressure change in any cross section of the tubing being revealed. It will allow to predict in advance the most dangerous sites from the point of view of long-life reliability of a column of pipes during cyclic liquid injection in the system of reservoir pressure maintenance.