A theory for the most stable variable viscosity profile in graded mobility displacement processes

Abstract

The stability of the displacement of a viscous fluid in a porous medium by a less viscous fluid containing an additive is considered. For the case in which the total amount of injected additive is fixed, a theory has been formulated to find the graded mobility profile which minimizes the instability due to an adverse mobility ratio. The result is a constrained nonlinear eigenvalue problem in which the mobility ratio of the fluids, a, and the total dimensionless amount of additive, N , appear as parameters. Estimates show N is typically large for practical situations. An asymptotic solution for large N is developed which shows that the optimal mobility profile is always an exponential curve at leading order. It is shown that although the optimal mobility profile is nearly exponential, the optimal concentration profile is not necesarily so, contrary to previous suggestions in the literature. Furthermore, it is shown that, unlike the case of a mobility jump, a graded mobility process under optimal conditions has amplifications independent of displacement velocity. We also discuss a possibile stabilization mechanism due to the very long wavelength disturbances which occur under optimal conditions.

An Instability Phenomenon in Hele-Shaw Displacements

“…An intermediate fluid with a variable viscosity between the displacing fluids can minimize the Saffman-Taylor instability when the surface tensions are different from zero -see the papers [15], [16], [17], [25], [26], [31].…”

“…An intermediate fluid with variable viscosity in a middle layer between the displacing fluids can minimize the Saffman-Taylor instability if surface tensions exist -see experimental and numerical results in [15], [16], [17], [25], [26], [31]. In [9], [10], [11] are given theoretical results concerning the linear stability of such three -layer Hele-Shaw flow.…”

Three-Layer Hele-Shaw Displacement With an Intermediate Non-Newtonian Fluid

“…Some previous results were obtained when the middle region is filled by a polymer solution with variable viscosity, which is function of the polymer concentration c, or is verifying a diffusion equation. In the first case -see [3], [6], [7], [10] -it was proved that an "optimal" viscosity profile exists, which can give a significant improvement of the stability. In the second case -see [4], [5], [11] -it was proved that the diffusion is improving the stability, compared with the Saffman-Taylor case, when the viscosity profile is verifying some specific conditions.…”

A paradox in Hele-Shaw displacements

“…minimize the Saffman-Taylor instability when the surface tensions are different from zerosee the papers [12], [13], [14], [15], [16], [17].…”