Miscible porous media displacements in the quarter five-spot configuration. Part 2. Effect of heterogeneities

Direct numerical simulations are employed to investigate the coupling between the viscous fingering instabilty and permeability heterogeneities for miscible displacements in quarter five-spot flows. Even moderate 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 viscous fingering instability. In contrast to their homogeneous counterparts (cf. Part 1, Chen & Meiburg 1998), heterogeneous quarter five-spot flows are seen to exhibit a more uniform dominant length scale throughout the entire flow domain. In line with earlier findings for unidirectional displacements, an optimal interaction of the mobility and permeability related vorticity modes can occur when the viscous length scale is of the same order as the correlation length of the heterogeneities. This resonance mechanism results in a minimal breakthrough recovery for intermediate correlation lengths, at fixed dimensionless flow rates in the form of a Peclet number Pe. However, for a constant correlation length, the recovery does not show a minimum as Pe is varied. Confirming earlier observations, the simulations show a more rapid breakthrough as the variance of the permeability variations increases. However, this tendency is far more noticeable in some parameter regimes than in others. It is furthermore observed that relatively low variances usually cannot change the tendency for a dominant finger to evolve along the inherently preferred diagonal direction, especially for relatively small correlation lengths. Only for higher variances, and for larger correlation lengths, are situations observed in which an off-diagonal finger can become dominant. Due to the nonlinear nature of the selection mechanisms at work, a change in the variance of the heterogeneities can result in the formation of dominant fingers along entirely different channels.