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Transient effects in oilfield cementing flows: Qualitative behaviour

  • M. A. MOYERS-GONZÁLEZ (a1), I. A. FRIGAARD (a2), O. SCHERZER (a3) and T.-P. TSAI (a4)

We present an unsteady Hele–Shaw model of the fluid–fluid displacements that take place during primary cementing of an oil well, focusing on the case where one Herschel–Bulkley fluid displaces another along a long uniform section of the annulus. Such unsteady models consist of an advection equation for a fluid concentration field coupled to a third-order non-linear PDE (Partial differential equation) for the stream function, with a free boundary at the boundary of regions of stagnant fluid. These models, although complex, are necessary for the study of interfacial instability and the effects of flow pulsation, and remain considerably simpler and more efficient than computationally solving three-dimensional Navier–Stokes type models. Using methods from gradient flows, we demonstrate that our unsteady evolution equation for the stream function has a unique solution. The solution is continuous with respect to variations in the model physical data and will decay exponentially to a steady-state distribution if the data do not change with time. In the event that density differences between the fluids are small and that the fluids have a yield stress, then if the flow rate is decreased suddenly to zero, the stream function (hence velocity) decays to zero in a finite time. We verify these decay properties, using a numerical solution. We then use the numerical solution to study the effects of pulsating the flow rate on a typical displacement.

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[1] F. Andreu , C. Ballester , V. Caselles & J. M. Mazón (2001) The Dirichlet problem for the total variation flow. J. Funct. Analy. 180, 347403.

[2] G. I. Barenblatt , V. M. Entov & V. M. Ryzhik (1990) Theory of Fluid Flows Through Natural Rocks. Kluwer Academic Publishers, New York.

[3] S. H. Bittleston J. Ferguson & I. A. Frigaard (2002) Mud removal and cement placement during primary cementing of an oil well; laminar non-Newtonian displacements in an eccentric annular Hele–Shaw cell. J.Eng.Math. 43, 229253.

[10] D. Gilbarg & N. S. Trudinger (1983) Elliptic Partial Differential Equations of Second Order. Springer-Verlag, New York.

[12] J. W. He & R. Glowinski (2000) Steady Bingham fluid flow in cylindrical pipes: a time dependent approach to the iterative solution. Numer. Linear Algebra Appl. 7 (6), 381428.

[16] S. Pelipenko & I. A. Frigaard (2004) Two-dimensional computational simulation of eccentric annular cementing displacements. IMA J. Appl. Math. 69, 557583.

[18] A. Tehrani , S. H. Bittleston & P. J. Long (1993) Flow instabilities during annular displacement of one non-Newtonian fluid by another. Exp. Fluids 14, 246256.

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European Journal of Applied Mathematics
  • ISSN: 0956-7925
  • EISSN: 1469-4425
  • URL: /core/journals/european-journal-of-applied-mathematics
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