Transient effects in oilfield cementing flows: Qualitative behaviour

Abstract

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 nonlinear 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.

Kinematic instabilities in two-layer eccentric annular flows, part 2: shear-thinning and yield-stress effects

“…However, in the case that we have a yield surface, additional compatibility conditions are also satisfied by the stream function perturbation there, which render the perturbation equations regular. Details regarding this are given in [14] and a similar example of such problems is found in the study of shear instabilities for visco-plastic fluids, where again the inclusion of a yield stress does not bring about continuous spectra, see e.g. [17].…”

“…The underlying HeleShaw model is derived in [14,15], see also Fig. 1 for a schematic of the geometry.…”

“…of the normal velocity, whereas (10) defines the jump in normal derivative of Ψ. Condition (10) comes from continuity of the pressure, see [14,1] for further explanation.…”

“…pulsation of a pump, (this latter possibility is not studied here). Therefore, in [14,15] we have used a modified HeleShaw approach in which the linear acceleration terms are retained in the momentum balance. Evidently, should a linear instability grow then eventually the nonlinear inertial terms are also important.…”

On the high frequency oscillatory tube flow of healthy human blood

“…(see (84)) was discretized using a Chebyshev collocation method (see, for example, [23,24,34]) and the resulting generalized eigenvalue problem was solved using the MATLAB routine eig. We note here that even though spectral methods have enjoyed much use for the discretisation of eigenvalue problems, it is well known in the case of secondorder spectral differentiation operators, for example, that only a fraction 2/π of the eigenvalues of the continuous problem can be approximated accurately [42,43].…”