Fingering phenomena in immiscible displacement in porous media flow

Pressure-driven displacement of one fluid by another in porous media has many applications, including oil production and the removal of soil contaminants. When the displacing fluid is less viscous, an idealized flat displacement front is ill-posed with respect to the growth of transverse perturbations (Taylor–Saffman instability). In models that include relative permeabilities, a macroscopic mixing of the fluids occurs in a rarefaction wave in a displacement region with a final discontinuity (shock wave) to the pure displaced fluid (Buckley–Leverett model). This discontinuity can also suffer from Taylor–Saffman instability. The instability is regularized by capillary effects, and a most unstable wavenumber can be found. Growth of these instabilities leads to fingering of the displacing fluid. In this work, we review the appropriate models and present a robust numerical approach to computing the most unstable wave numbers for given parameters (provided they lead to a non-degenerate parabolic problem). This approach to quantitatively accurate solutions will be of use in fitting experimental results and evaluating models used in a number of applications. We show that the Buckley–Leverett shock can be stabilized due to relative permeability effects in some cases. We give a clear presentation of asymptotic results in the limit of small capillarity for this problem. These asymptotic results lead to far-field pressure conditions that greatly simplify the numerical computations. They also highlight some open mathematical questions in this area.