Flow in annular space occurs in drilling operation of oil and gas wells. The correct prediction of the flow of the drilling mud in the annular space between the well wall and the drill pipe is essential to determine the variation in the mud pressure within the wellbore, the frictional pressure drop, and the efficiency of the transport of the rock drill cuttings. A complete analysis of this situation is extremely complex: the inner cylinder is usually rotating, the wellbore wall will depart significantly from cylindrical, the drill pipe is eccentric, and the eccentricity varies along the well. A complete analysis of this situation would require the solution of the three-dimensional momentum equation and would be computationally expensive and complex. Models available in the literature to study this situation do consider the rotation of the inner cylinder and the non-Newtonian behavior of the drilling fluids, but assume the relative position of the inner with respect to the outer cylinders fixed, i.e., they neglect the variation of the eccentricity along the length of the well, and the flow is considered to be fully developed. This approximation leads to a two-dimensional model to determine the three components of the velocity field in a cross-section of the annulus. The model presented in this work takes into account the variation of the eccentricity along the well; a more appropriate description of the geometric configuration of directional wells. As a consequence, the velocity field varies along the well length and the resulting flow model is three-dimensional. Lubrication theory is used to simplify the governing equations into a two-dimensional differential equation that describes the pressure field. The results show the effect of the variation of the eccentricity on the friction factor, maximum and minimum axial velocity in each cross section, and the presence of azimuthal flow even when the inner cylinder is not rotating.

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