A Method for Designing Graded Viscosity Banks
Authors: Claridge E.L. 
Published: December 2007

Abstract
A method for designing graded banks, to prevent deleterious effects of viscous fingering, was obtained by repeated application of Koval's equations, which define the dimensionless length of a fingering zone. Examples are given for miscible and polymer flooding.

Introduction
While it has been shown experimentally that a gradual change from a more to a less viscous injected fluid (i.e., a graded bank) can mitigate the deleterious effects of viscous fingering on areal sweep efficiency in reservoir displacement processes, it also is apparent that the rate of processes, it also is apparent that the rate of change is important. Viscous fingering occurs despite the diffusional blending of fluids occurring at the surface of the fingers. The viscous fingering process follows the stretching behavior (linearly process follows the stretching behavior (linearly growing with distance or time) of the characteristic solutions of hyperbolic, partial differential equations, while the diffusion process grows with the square root of distance or time, reflecting the parabolic nature of the partial differential equation parabolic nature of the partial differential equation describing it. Thus, viscous fingers can grow faster than the diffusion band that tends to moderate them.
Attempts were made to define this competition using perturbation analysis. Chuoke's analysis for immiscible fingering was accompanied by research on miscible fluids available to me from private communication. Perrine published a private communication. Perrine published a similar linear analysis for miscible fluids that was challenged by Outmans, who published an analysis proceeding from linear (first-order) terms up to fourth-order terms. More terms indicated a more bulbous shape of developing fingers; but a relationship for the minimum wave length of perturbations that would not be eliminated by perturbations that would not be eliminated by transverse dispersion could be derived only from the linear form. Both Chuoke and Perrine obtained an equation relating a concentration gradient or viscosity gradient to this critical wave length, which enables predictions to be made for a graded bank. Chuoke's form of this equation is
C = 2 [DT /u(d /dx)] 1/2..................(1)
Fingers will be eliminated when C is equal to twice the maximum transverse breadth b of a linear displacement system. Substituting 2b and solving for the viscosity gradient,
d 1n /dx = DT 2/ub 2.........................(2)
Perrine obtained this equation in the form Perrine obtained this equation in the formdc/dx = DT 2/ub2(dln /dc).....................(3)
If we multiply both sides of Perrine's equation by (d ln /dC), we get Eq. 2. Integrating,
ln (2/ 1) = ln M = x DT 2/ub2...........(4)
As an example, let b = 660 ft (width of a 1:1 line drive with 10 acres/well), D T = 1 x 10(-4) sq cm/sec; u = 3.528 x 10(-4) cm/sec (1 ft/D); and M = 10. We obtain x = 11 million ft (about 16,500 PV through-put) as the length required to make this mobility ratio change to avoid completely any viscous fingering, which seems unattainable in practice.
Kyle and Perrine performed experiments in a 9-ft-long sand pack that explains this further. From their work, I conclude thatTo make a mobility change of 4.85:1 a graded bank length of 295 ft (33 PV) is calculated using the above equation. When this change was made in shorter (5-, 10-, or 20-ft) graded banks, viscous fingers were observed, but the fingers were successively weaker as the gradient decreased. When a graded bank 50 ft long was used, fingers did not appeal within the 9-ft length of the pack. They might have become visible had a longer pack been used.When fingers were observed, their wave length in the transverse direction was in accordance with predictions made with computer simulations based predictions made with computer simulations based on first- and second-order effects. The shapes of fingers were reasonably in accord with the simulation, although the simulated fingers were fuzzy. The fingering-zone length was not appreciably affected by displacement rate.
SPEJ
P. 315