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

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, α, 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.