Numerical investigation of viscous fingering in a three-dimensional cubical domain

We perform three-dimensional numerical simulations to understand the role of viscous fingering in sweeping a high-viscous fluid (HVF). These fingers form due to the injection of a low-viscous fluid (LVF) into a porous media containing the high-viscous fluid. We find that the sweeping of HVF depends on different parameters such as the Reynolds number ($Re$) based on the inflow rate of the LVF, the P\'eclet number ($Pe$), and the logarithmic viscosity ratio of HVF and LVF, $\mathfrak{R}$. At high values of $Re$, $Pe$, and $\mathfrak{R}$, the fingers grow non-linearly, resulting in earlier tip splitting of the fingers and breakthrough, further leading to poor sweeping of the HVF. In contrast, the fingers evolve uniformly at low values of $Re$, $Pe$, and $\mathfrak{R}$, resulting in an efficient sweeping of the HVF. We also estimate the sweep efficiency and conclude that the parameters $Re$, $Pe$ and $\mathfrak{R}$ be chosen optimally to minimize the non-linear growth of the fingers to achieve an efficient sweeping of the HVF.