Numerical simulations are presented which, in conjunction with the accompanying experimental investigation by Petitjeans & Maxworthy (1996), are intended to elucidate the miscible flow that is generated if a fluid of given viscosity and density displaces a second fluid of different such properties in a capillary tube or plane channel. The global features of the flow, such as the fraction of the displaced fluid left behind on the tube walls, are largely controlled by dimensionless quantities in the form of a Péclet number Pe, an Atwood number At, and a gravity parameter. However, further dimensionless parameters that arise from the dependence on the concentration of various physical properties, such as viscosity and the diffusion coefficient, result in significant effects as well.The simulations identify two distinct Pe regimes, separated by a transitional region. For large values of Pe, typically above O(10), a quasi-steady finger forms, which persists for a time of O(Pe) before it starts to decay, and Poiseuille flow and Taylor dispersion are approached asymptotically. Depending on the strength of the gravitational forces, we observe a variety of topologically different streamline patterns, among them some that leak fluid from the finger tip and others with toroidal recirculation regions inside the finger. Simulations that account for the experimentally observed dependence of the diffusion coefficient on the concentration show the evolution of fingers that combine steep external concentration layers with smooth concentration fields on the inside. In the small-Pe regime, the flow decays from the start and asymptotically reaches Taylor dispersion after a time of O(Pe).An attempt was made to evaluate the importance of the Korteweg stresses and the consequences of assuming a divergence-free velocity field. Scaling arguments indicate that these effects should be strongest when steep concentration fronts exist, i.e. at large values of Pe and At. However, when compared to the viscous stresses, Korteweg stresses may be relatively more important at lower values of these parameters, and we cannot exclude the possibility that minor discrepancies observed between simulations and experiments in these parameter regimes are partially due to these extra stresses.

A detailed two-part computational investigation is conducted into the dynamical evolution of two-dimensional miscible porous media flows in the quarter five-spot arrangement of injection and production wells. High-accuracy direct numerical simulations are performed that reproduce all dynamically relevant length scales in solving the vorticity–streamfunction formulation of Darcy's law. The accuracy of the method is demonstrated by a comparison of simulation data with linear stability results for radial source flow.Within this part, Part 1 of the present investigation, a series of simulations is discussed that demonstrate how the mobility ratio and the dimensionless flow rate denoted by the Péclet number Pe affect both local and integral features of homogeneous displacement processes. Mobility ratios up to 150 and Pe-values up to 2000 are investigated. For sufficiently large Pe-values, the flow near the injection well gives rise to a vigorous viscous fingering instability. As the unstable concentration front approaches the central region of the domain, nonlinear interactions between the fingers similar to those known from unidirectional flows are observed, such as merging, partial merging, and shielding, along with secondary tip-splitting and side-branching instabilities. At large Pe-values, several of these fingers compete for long times, before one of them accelerates ahead of the others and leads to the breakthrough of the front.In contrast to unidirectional flows, the quarter five-spot geometry imposes both an external length scale and a time scale on the flow. The resulting spatial non-uniformity of the potential base flow is observed to lead to a clear separation in space and time of large and small scales in the flow. Small scales occur predominantly during the early stages near the injection well, and at late times near the production well. The central domain is dominated by larger scales.Taken together, the results of the simulations demonstrate that both the mobility ratio and Pe strongly affect the dynamics of the flow. While some integral measures, such as the recovery at breakthrough, may show only a weak dependence on Pe for large Pe-values, the local fingering dynamics continue to change with Pe.The increased susceptibility of the flow to perturbations during the early stages provides the motivation to formulate an optimization problem that attempts to maximize recovery, for a constant overall process time, by employing a time-dependent flow rate. Within the present framework, which accounts for molecular diffusion but not for velocity-dependent dispersion, simulation results indeed indicate the potential to increase recovery by reducing the flow rate at early times, and then increasing it during the later stages.

The effect of geometry is observable on the onset of instability, where we obtain significant differences from existing results in the rectilinear geometry.

Direct numerical simulations are employed to investigate the coupling between the viscous fingering instability and permeability heterogeneities for miscible displacements in quarter five-spot flows. Even moderate inhomogeneities are seen to have a strong effect on the flow, 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 Péclet 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.

Miscible quarter five-spot displacements in a Hele-Shaw cell are investigated by means of experimental measurements and numerical simulations. The experiments record both the volumetric as well as the surface efficiency at breakthrough as a function of the dimensionless flow rate in the form of a Peclet number and the viscosity contrast. For small flow rates, both of these efficiency measures decrease uniformly with increasing Peclet numbers. At large flow rates, an asymptotic state is reached where the efficiencies no longer depend on the Peclet number. Up to Atwood numbers of approximately 0.5, the less viscous fluid occupies close to 23 of the gap width, which indicates a near-parabolic velocity profile across the gap. Consequently, in this parameter range a Taylor dispersion approach should be well suited to account for flow-induced dispersion effects. For larger viscosity contrasts, accompanying two-dimensional numerical simulations based on Taylor dispersion predict an increased stabilization for high flow rates, which is not confirmed by the experiments. This suggests that, in order to extend the range of applicability to larger viscosity contrasts, the components of the dispersion tensor will have to be amended in order to account for the presence of quasi-steady fingers.