Cy. Chen et E. Meiburg, MISCIBLE POROUS-MEDIA DISPLACEMENTS IN THE QUARTER 5-SPOT CONFIGURATION - PART 2 - EFFECT OF HETEROGENEITIES, Journal of Fluid Mechanics, 371, 1998, pp. 269-299
Direct numerical simulations are employed to investigate the coupling
between the viscous fingering instabilty and permeability heterogeneit
ies for miscible displacements in quarter five-spot flows. Even modera
te inhomogeneities are seen to have a strong effect on the how, which
can result in a complete bypass of the linear growth phase of the visc
ous fingering instability. In contrast to their homogeneous counterpar
ts (cf. Part 1, Chen & Meiburg 1998), heterogeneous quarter five-spot
flows are seen to exhibit a more uniform dominant length scale through
out the entire flow domain. In line with earlier findings for unidirec
tional displacements, an optimal interaction of the mobility and perme
ability related vorticity modes can occur when the viscous length scal
e is of the same order as the correlation length of the heterogeneitie
s. This resonance mechanism results in a minimal breakthrough recovery
for intermediate correlation lengths, at fixed dimensionless flow rat
es in the form of a Peclet number Pe. However, for a constant correlat
ion length, the recovery does not show a minimum as Pe is varied. Conf
irming earlier observations, the simulations show a more rapid breakth
rough as the variance of the permeability variations increases. Howeve
r, this tendency is far more noticeable in some parameter regimes than
in others. It is furthermore observed that relatively low variances u
sually cannot change the tendency for a dominant finger to evolve alon
g the inherently preferred diagonal direction, especially for relative
ly small correlation lengths. Only for higher variances, and for large
r correlation lengths, are situations observed in which an off-diagona
l finger can become dominant. Due to the nonlinear nature of the selec
tion mechanisms at work, a change in the variance of the heterogeneiti
es can result in the formation of dominant fingers along entirely diff
erent channels.