Simulation of flexural waves in drill pipes including the effects of the gravitational field

We design a full-wave modeling method to simulate flexural (bending) waves in pipes of variable cross-section and material properties, subject to the effects of the gravitational field. The study finds application in propagation through drill strings and stability of hydrocarbon wells while drilling. The algorithm is based on a direct (grid) method, where the spatial derivatives are computed with the Fourier pseudospectral method. The modeling is successfully tested against the analytical solution for flexural waves propagating in a uniform pipe. Moreover, we obtain reciprocity relations for the lateral deflection, particle velocity and bending moment due to sources applied in the force-balance and constitutive equations. The relations are then verified by the simulations and at the same time provide a test of the consistency of the numerical algorithm in inhomogeneous media, where there is no closed-form analytical solution.We analyze the effects of the gravitational field and show that for negative axial loads (compressive stresses) there is an instability below a critical wavenumber, while the system is stable for positive (tensile) loads. Such a situation appears below the neutral point of a drill string when the gravitational force is taken into account. A plane-wave analysis yields the phase and group velocities when the load is uniform along the pipe and in the case of a suspended pipe subject to a gravitational force with a neutral point downhole. In the latter case, the signal propagates with attenuation. Propagation along a realistic drill string, simulating the drill bit as a stress-free end, is illustrated. Moreover, propagation subject to an axial gravitational load shows that the signal is slower for a tensile prestress.