Miscible porous media displacements in the quarter five-spot configuration. Part 1. The homogeneous case

Abstract

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.