HOC simulations of miscible viscous fingering of a finite slice: A new insight
We investigate the dynamics of viscous fingering (VF) in miscible slices in homogeneous, isotropic porous media. The fluid flow is governed by incompressible Darcy's law, whereas the solute transport is described using an advection-diffusion equation. The viscosity of the miscible system depends on the solute concentration, creating a viscosity contrast between the displacing fluid and the finite sample. When expressed in terms of stream function, the flow is described by a system of nonlinear, two-way coupled advection-diffusion type equations. We consider three types of boundary conditions: (a) periodic, (b) impermeable (zero normal velocity) and no-flux (solute), and (c) permeable (allowing non-zero normal velocity) and no diffusive flux (solute) transverse boundaries. This initial boundary value problem is solved numerically using a fourth-order compact finite difference method, while the Crank-Nicolson technique is used for time integration. Although the onset of viscous fingering and early time behavior are independent of the choice of boundary types, long-time behavior, solute mixing and spreading depend on the boundary conditions. In particular, it is observed that the permeable boundaries allow solute mass to increase, leading to stronger fingering instabilities, larger mixing lengths and non-trivial evolution of interfacial lengths. The findings of this study have implications in chromatography separation.