Open Access Peer-reviewed

Numerical Solution of Power-law Fluid Flow through Eccentric Annular Geometry

Nuha Hussein Ebrahim1,, Noaman El-Khatib1, Mariyamni Awang1

1Petroluem Engineering, Universiti Teknologi Petronas, Perak Darul Ridzuan, Malaysia

American Journal of Numerical Analysis. 2013, 1(1), 1-7. DOI: 10.12691/ajna-1-1-1
Published online: August 25, 2017


Cuttings transport modeling in inclined and horizontal wellbores is complicated due to the eccentricity of the annulus. Development of a model for cuttings transport requires a deep understanding of the drilling mud flow behavior in the eccentric annular geometry. In this paper, bipolar coordinates system is used to solve for the eccentric annular geometry due to irregular shape of the boundaries. Finite difference method is used to obtain the velocity profile of Power-law non-Newtonian fluids through eccentric annuli. The discretized dimensionless Equation of flow using the finite difference method is solved iteratively using Point Successive Over Relaxation (S.O.R.) method. The results for Newtonian eccentric annular flow with 0.0001 eccentricity are in agreement with the Newtonian concentric annular flow. The Power-law eccentric annular flow results with flow index of 1.0 are verified with Newtonian fluid eccentric annular flow results. The parametric effects of flow index, Pipe/hole radius ratio, and eccentricity are investigated. We expect the development of a new model for flow in eccentric annular geometries to be an important new tool for application in oil - well drilling and production.


bipolar coordinate, eccentric annulus, Power-law fluid, Point Successive over Relaxation
[1]  Chin, W., Computational rheology for pipelines and annular flow, Gulf Publishing Company, Houston, Texasm, 2001.
[2]  Alderman, N. Ram Babu, D. Hughes, T. and Maitland, G. “The rheological properties of oilwell drilling fluids,” Proc Xth Int. Cong. Rheology, Sydney, 140-142, 1998.
[3]  Hamed, S. and Belhadri, M. “Rheological properties of biopolymers drilling fluids,” Journal of Petroleum Science and Engineering, 67 (3-4), 84-90, 2009.View Article
[4]  Carden, R. and Grace, R., Horizontal and directional drilling, PetroSkills. OGCI Co., Oklahoma, USA, 2007. PubMed
[5]  J. Heyda, “A green’s function solution for the case of laminar incompressible flow between non-concentric circular cylinders,” Journal of Franklin Inst. 267, 25-34, 1959.View Article
[6]  Redberger, P. and Charles, M., “Axial laminar flow in a circular pipe containing a fixed eccentric core,” Canadian J. of Chem. Eng., 40(4), 148-151, 1962.View Article
[7]  Snyder, W. and Goldstein, G., “An Analysis of Fully Developed Laminar Flow in an Eccentric Annuli,” AIChE Journal, 11, 462-467, 1965.View Article
[8]  Mitsuishi, N. and Aoyagi, Y., “Non-Newtonian Fluid Flow in an Eccentric Annulus,” Journal of Chemical Eng., 6(5), 402-08, 1973.
[9]  Guckes, T.,“Laminar flow of non-Newtonian Fluids in an eccentric annulus,” Trans. ASME, 97(2), 498-506, 1974.
[10]  Tosun, I., “Axial laminar Flow in an Eccentric Annulus: An Approximate Solution,” AICHE Journal, 30(5), 877-78, 1984.View Article
[11]  Bird, R. Stewart, W. and Lightfoot, E., Transport Phenomena, John Wiley & Sons, New York City, 1960
[12]  Haciislamoglu, M. and Langlinais, J., “Non-Newtonian Flow in Eccentric Annuli,” J. Energy Res. Tech. 112(3), 163-169, 1990.View Article
[13]  Tao, L. and Donovan, W., “Through Flow in Concentric and Eccentric Annuli of Fine Clearance With and Without Relative Motion of the Boundaries,” Trans., ASME, 77, 1291-1301, 1995.
[14]  Vaughn, R., “Axial Laminar flow of Non-Newtonian Fluids in Narrow Eccentric Annuli,” SPEJ, 277-80, Trans., AIME, 234, 1965.
[15]  Iyoho, A. and Azar, J., “An Accurate Slot Model for Non-Newtonian Flow through Eccentric Annuli,” SPEJ (Oct. 1981) 565-72, 1981.
[16]  Uner, D. Ozden, C., and Tosun, I., “An Approximate Solution for Non-Newtonian Flow in Eccentric Annuli,” Ind. And eng. Chem., 27(4), 698-701, 1988.
[17]  Luo, Y. and Peden, J., “Flow of Non-Newtonian Fluids Through Eccentric Annuli,” SPE Production Engineering, 91-96, 1990.
[18]  Pillutla, J., “Laminar non-Newtonian flows in eccentric annuli with inner cylinder rotation,” Msc diss., University of Indhra, 2001.
[19]  Speigel, M. R., Mathematical Handbook of Formulas and Tables, McGraw-Hill Book Co., pp. 125-128, 1968.
[20]  Haciislamoglu, M. and Langlinais, J., “Effect of pipe Eccentricity on Surge Pressures,” Journal of Energy Resources Technology, 113, 157-160, 1991.View Article
[21]  Adariani, Y. H.: “Numerical simulation of laminar flow of non-Newtonian fluids in eccentric annuli,” Msc diss., University of Tulsa, 2005.
[22]  Nouri, J.M., Whitelaw, J.H., “Flow of Newtonian and non-Newtonian fluids in a concentric annulus with rotation of the inner cylinder,” Journal of Fluids Engineering, 116, 1994, 821-827.View Article