HOC simulations of miscible viscous fingering of a finite slice: A new insight

TL;DR

This study uses high-order simulations to analyze viscous fingering in miscible slices within porous media, revealing how boundary conditions influence long-term behavior, mixing, and instability growth.

Contribution

It introduces a numerical approach combining fourth-order finite differences and Crank-Nicolson for modeling viscous fingering with various boundary conditions in miscible systems.

Findings

Permeable boundaries lead to increased solute mass and stronger fingering.

Long-time behavior depends on boundary conditions, affecting mixing and interface evolution.

Initial fingering onset is independent of boundary types.

Abstract

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.