Inertial effects during irreversible meniscus reconfiguration in angular pores

In porous media, the dynamics of the invading front between two immiscible fluids is often characterized by abrupt reconfigurations caused by local instabilities of the interface. As a prototype of these phenomena we consider the dynamics of a meniscus in a corner as it can be encountered in angular pores. We investigate this process in detail by means of direct numerical simulations that solve the Navier-Stokes equations in the pore space and employ the Volume of Fluid method (VOF) to track the evolution of the interface. We show that for a quasi-static displacement, the numerically calculated surface energy agrees well with the analytical solutions that we have derived for pores with circular and square cross sections. However, the spontaneous reconfigurations are irreversible and cannot be controlled by the injection rate: they are characterized by the amount of surface energy that is spontaneously released and transformed into kinetic energy. The resulting local velocities can be orders of magnitude larger than the injection velocity and they induce damped oscillations of the interface that possess their own time scales and depend only on fluid properties and pore geometry. In complex media (we consider a network of cubic pores) reconfigurations are so frequent and oscillations last long enough that increasing inertial effects leads to a different fluid distribution by influencing the selection of the next pore to be invaded. This calls into question simple pore-filling rules based only on capillary forces. Also, we demonstrate that inertial effects during irreversible reconfigurations can influence the work done by the external forces that is related to the pressure drop in Darcy's law. This suggests that these phenomena have to be considered when upscaling multiphase flow because local oscillations of the menisci affect macroscopic quantities and modify the constitutive relationships to be used in macro-scale models. These results can be extrapolated to other interface instabilities that are at the origin of fast pore-scale events, such as Haines jumps, snap-off and coalescence.