The article deals with the phenomenon of hydraulic hammer when the pumps are stopped in the final stage of hydraulic fracturing. The authors propose a mathematical model of a fracture and a well, which are coupled at the perforation point. Both objects are considered as one-dimensional waveguides in order to take into account the possible oscillatory nature of the pressure disturbance propagation. The fracture model is considered within the Perkins – Kern – Nordgren (PKN) approach and augmented with inertial terms in the equations. The coupled model allows to obtain the pressure signal at different points in the well, taking into account the influence of the fracture. The simplified formulation of the problem, taking into account the linearisation of the equations and the absence of fluid leakage, allows to obtain analytical solutions for the fracture and the well separately. The authors discuss the principal differences between the fracture and the well as waveguides. In particular, the propagation of perturbations in the well is determined by the compressibility of the fluid and in the fracture by the elasticity of the walls. This results in a high reflection coefficient at the perforation point, so that only a small fraction of the oscillation energy is transferred from the well to the fracture and back. In addition to the oscillatory mode of perturbation propagation in the waveguide, the viscous pressure relaxation mode is possible, in which there are no oscillations. In this case, the viscosity of the fluid is the most important parameter. The oscillatory mode is only realised in the fracture when low viscosity fluid up to 30 mPa∙s is injected. In current hydraulic fracturing technology, the last stage of injection is performed on a linear gel, so the oscillatory mode is almost always realised in the well. The coupled modelling of the fracture and the wellbore is performed numerically using the control volume method. The total signal has fracture and borehole components which differ in their frequencies. The propagation of oscillations in a variable cross section fracture is discussed. In the PKN model, the crack width reduces towards the tip. In this case a combined mode of disturbance propagation is realised. This implies oscillations in the initial part of the crack and viscous relaxation in the remaining part. This leads to a change in the parameters of the fracture as a waveguide not only along its length but also in time.
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