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Ind. Eng. Chem. Res. 1995,34, 936-942
Couette Flow of Non-Newtonian Power-Law Fluids in Narrow Eccentric Annuli Liao Yang and Godwin A Chukwu" Department of Petroleum Engineering, University of Alaska Fairbanks, 437 Duckering Building, Fairbanks, Alaska 99775-5880
The analysis of the steady laminar Couette flow of non-Newtonian power-law fluids in a narrow eccentric annulus is employed in this study to compute the surge or swab pressure encountered when running or pulling tubular goods in a liquid-filled borehole, respectively. Excessive surge pressure can fracture the formation, while uncontrolled swab pressure can result in well blowout. In this study, the eqs of motion are analytically solved and the solution of these eqs is presented in both dimensionless and graphical forms for a more general application to computing the surge or swab pressure. The family of curves is presented for different pipehorehole eccentricity ratios and power-law fluid index values which span the range of typical drilling fluids. By employing the computed surge pressures, in combination with the family of curves, the maximum velocity a t which the casing can be run in the hole without the danger of fracturing the formation can be obtained. The expected error in surge computation for a narrow concentric annulus represented by a slot, as a result of eccentricity, is evaluated. The results obtained from the these analyses will aid in proper design and optimization of drilling programs, especially in deviated holes.
Introduction Fluid flow phenomenon whereby the fluid is confined between two coaxial cylinders, one of which is stationary and the other is moving at a uniform velocity, is known as Couette flow. This flow characteristic is representative of flow in the wellbore annulus where the wall of the wellbore is represented by the stationary cylinder, and the drill string or casing pipe is represented by the stationary cylinder. The fluid average velocity is dependent on the velocity of the moving cylinder or pipe. The displacement of drilling fluid by the movement of the drill string or casing produces pressure variations in the borehole. The velocity of this displacement governs the magnitude of the pressure change which can increase or decrease the hydrostatic pressure of the drilling fluid, to produce surge or swab pressure, respectively. There has been a basic assumption in the petroleum industry that the drill string or casing pipes are concentrically positioned in the borehole during drilling operations. Many fluid flow models have been developed based on this assumption. Not until this age of both directional and horizontal drilling technology have researchers and engineers alike found out that the pressure difference between the two borehole conditions has a great impact on the overall drilling and well performance. Early studies by Cardwell (19531, Cannon (19341, Horn (19501, Goins et al. (1951), Clark (19551, Burkhardt (19611, Schuh (19641, McEachern (1966),Lubinski et al. (19771, La1 (19831, and Mitchell (1988) had all assumed a concentric annular geometry, and in most cases, the surge or swab pressure predicted was based on fluid average velocity instead of pipe velocity. Both the analytical and numerical solutions of non-Newtonian fluid flow through eccentric annuli have been presented by several authors (Vaughn, 1965; Guckes, 1975; Iyoho, 1981; Luo and Peden, 1987; Ozgen and Tosun, 1987). Walton and Bittleston (1991) considered the specialized
case of Bingham plastic fluid in a narrow eccentric annulus. Only Haciislamaglu and Langlinais (1991) had considered the numerical solution of non-Newtonian fluids in the important case of a moving inner cylinder or pipe. Most recent work by Chukwu and Blick (1991) on the Couette flow of non-Newtonian power-law fluids was the development of a graphical technique of computing the surge or swab pressure in a concentric annular geometry. Redberger and Charles (19621, using a finite difference method, calculated the volumetric flow rate in an eccentric annulus and showed that the displacement of the inner pipe from a concentric position increased the volumetric flow rate for a given pressure gradient. This study is an attempt to analyze the annular flow of non-Newtonian power-law fluids based on the eccentricity of the moving inner drill string or casing pipe in the borehole. This will entail an analytical solution of the eqs of motion and presenting the solution in a nondimensional and less complex form, for a more general application, for computing the surge or swab pressure encountered when running tubular goods in a liquid-filled borehole.
Governing Equations The eq of motion for a steady, laminar incompressible fluid flow through an eccentric annulus can be expressed in cylindrical coordinates as
1a r ar
*
- -(rrrz)
aGz + ap* + -r1 =0 a0 az
where P* is the dynamic pressure and z is measured axially in the direction of flow. The rheological model that is used to represent power-law fluids in cylindrical coordinates can be written as
* Corresponding author.E-mail:
[email protected] 0888-5885/95l2634-0936$09.00/0
0 1995 American Chemical Society
Ind. Eng. Chem. Res., Vol. 34, No. 3,1995 937
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