Cy. Chen et E. Meiburg, MISCIBLE POROUS-MEDIA DISPLACEMENTS IN THE QUARTER 5-SPOT CONFIGURATION - PART 1 - THE HOMOGENEOUS CASE, Journal of Fluid Mechanics, 371, 1998, pp. 233-268
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. H
igh-accuary direct numerical simulations are performed that reproduce
all dynamically relevant length scales in solving the vorticity-stream
function formulation of Darcy's law. The accuracy of the method is dem
onstrated by a comparison of simulation data with linear stability res
ults for radial source flow. Within this part, Part 1 of the present i
nvestigation, a series of simulations is discussed that demonstrate ho
w the mobility ratio and the dimensionless flow rate denoted by the Pe
clet 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 instabi
lity. As the unstable concentration front approaches the central regio
n 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 fi
ngers 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 e
xternal length scale and a time scale on the flow. The resulting spati
al 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 fl
ow. Small scales occur predominantly during the early stages near the
injection well, and at late times near the production well. The centra
l domain is dominated by larger scales. Taken together, the results of
the simulations demonstrate that both the mobility ratio and Pe stron
gly affect the dynamics of the flow. While some integral measures, suc
h as the recovery at breakthrough, may show only a weak dependence on
Pe for large Pe-values, the local fingering dynamics continue to chang
e with Pe. The increased susceptibility of the flow to perturbations d
uring the early stages provides the motivation to formulate an optimiz
ation problem that attempts to maximize recovery, for a constant overa
ll process time, by employing a time-dependent flow rate. Within the p
resent framework, which accounts for molecular diffusion but not for v
elocity-dependent dispersion, simulation results indeed indicate the p
otential to increase recovery by reducing the flow rate at early times
, and then increasing it during the later stages.