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TECHNICAL PAPERS

Three-Dimensional Flow of a Newtonian Liquid Through an Annular Space with Axially Varying Eccentricity

[+] Author and Article Information
Eduarda P. de Pina

Department of Mechanical Engineering, Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rua Marquês de São Vicente, 225 Gavea, 22493-900, Rio de Janeiro, RJ, Brazil

M. S. Carvalho

Department of Mechanical Engineering, Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio), Rua Marquês de São Vicente, 225 Gavea, 22493-900, Rio de Janeiro, RJ, Brazilmsc@mec.puc-rio.br

J. Fluids Eng 128(2), 223-231 (Aug 25, 2005) (9 pages) doi:10.1115/1.2170126 History: Received April 12, 2005; Revised August 25, 2005

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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Copyright © 2006 by American Society of Mechanical Engineers
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Figures

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Figure 1

Configuration of annular space with eccentricity varying along the axial direction

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Figure 2

Domain where the pressure equation is solved. The height of the flow channel varies with the axial and azimuthal coordinates.

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Figure 3

f×Re as a function of eccentricity parameter at κ=0.2, κ=0.5, and κ=0.8 predicted by the lubrication model and the two-dimensional solution presented by Escudier (8)

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Figure 4

Axial velocity u∕U¯ at eccentricity ε=e∕(R−Ri)=0, ε=0.33, and ε=0.98. The radius ratio is k=0.5.

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Figure 5

Axial velocity u∕U¯ at eccentricity ε=e∕(R−Ri)=0, ε=0.33, and ε=0.98. The radius ratio is k=0.8.

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Figure 6

Maximum velocity umax∕U¯ as a function of the eccentricity parameter

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Figure 7

Sketch of annular space with the position of the center of the inner cylinder described by a sinusoidal function

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Figure 8

(a) Axial and azimuthal velocity fields. The maximum axial velocity is umax∕U¯=2.91. (b) Pressue field. The axial position is z∕L=0.25, the amplitude of the sinusoidal variation of the eccentricity is A∕(R0−Ri)=0.98, and the wavelength is λ∕R0=164.

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Figure 9

(a) Axial and azimuthal velocity field. The maximum axial velocity is umax∕U¯=2.89. (b) Pressue field. The axial position is z∕L=0.35, the amplitude of the sinusoidal variation of the eccentricity is A∕(R0−Ri)=0.98, and the wavelength is λ∕R0=164.

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Figure 10

(a) Axial and azimuthal velocity fields. The maximum axial velocity is umax∕U¯=1.6. (b) Pressue field. The axial position is z∕L=0.5, the amplitude of the sinusoidal variation of the eccentricity is A∕(R0−Ri)=0.98, and the wavelength is λ∕R0=164.

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Figure 11

Friction factor as a function of the amplitude of the sinusoidal variation of the eccentricity along the axial direction

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Figure 12

Friction factor as a function of the wavelength of the sinusoidal variation of the eccentricity along the axial direction

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Figure 13

Dimensionless pressure in the flow channel (a) and dimensionless channel height (b), which are a function of the axial and azimuthal coordinate

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