Propagation model with multi-boundary conditions for periodic mud pressure wave in long wellbore

Periodic pressure waves propagating in drilling fluids along the wellbore are widely used to transmit the downhole information through the drillstring. Although various evaluation models for the wellbore and periodic pipe flows have been studied in published literature, the influence of the frequency and multi-boundary conditions on these models for the non-Newtonian drilling fluid is rarely considered. The paper presents a one-dimensional equivalent model to simulate the propagation of the periodic pressure waves in drilling fluids along the wellbore based on the assumption that the system is isothermal and the internal diameter of drillstring is a constant value. Specially, the periodic pressure waves are divided into low-frequency part (such as pump-on pulse) and high-frequency part (such as pressure wave generated by mud pulse telemetry system). The interactions of these two parts with the turbulence and Bingham fluid are investigated based on the Navier–Stokes equation for a 2D axisymmetric transient flow subjected to the five boundary conditions including the pump, air chamber, mud pulse telemetry system, drill bit and the outlet. A finite difference method with second-order accuracy is used to solve the numerical model. Experiments in the shallow and deep wells are conducted to validate the model. The simulated results are in good agreement with the experiment measurement. Finally, the influence of the frequency on pressure wave attenuation is analyzed and the result is compared with those calculated from acoustic attenuation formula. Both results show consistent trend with increasing frequency.