A detailed two-part computational investigation is conducted into
evolution of two-dimensional miscible porous media flows in the quarter
arrangement of injection and production wells. High-accuracy direct numerical
are performed that reproduce all dynamically relevant length scales in
the vorticity–streamfunction formulation of Darcy's law. The
accuracy of the method
is demonstrated by a comparison of simulation data with linear stability
radial source flow.
Within this part, Part 1 of the present investigation, a series of simulations
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
the central region of the domain, nonlinear interactions between the fingers
those known from unidirectional flows are observed, such as merging, partial
and shielding, along with secondary tip-splitting and side-branching instabilities.
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
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
early stages near the injection well, and at late times near the production
central domain is dominated by larger scales.
Taken together, the results of the simulations demonstrate that both
ratio and Pe strongly affect the dynamics of the flow. While some
such as the recovery at breakthrough, may show only a weak dependence on
large Pe-values, the local fingering dynamics continue to change
The increased susceptibility of the flow to perturbations during the
provides the motivation to formulate an optimization problem that attempts
maximize recovery, for a constant overall process time, by employing a
flow rate. Within the present framework, which accounts for molecular diffusion
not for velocity-dependent dispersion, simulation results indeed indicate
to increase recovery by reducing the flow rate at early times, and then
during the later stages.