Disorder-induced non-linear growth of viscously-unstable immiscible two-phase flow fingers in porous media

The immiscible displacement of a fluid by another one inside a porous medium produces different types of patterns depending on the capillary number Ca and viscosity ratio M. At high Ca, viscous fingers resulting from the viscous instability between fluid-fluid interfaces are believed to exhibit the same Laplacian growth behavior as viscously-unstable fingers observed in Hele-Shaw cells by Saffman and Taylor [1], or as diffusion limited aggregates (DLA) [2]. I.e., the interface velocity depends linearly on the local gradient of the physical field that drives the growth process (for two-phase flow, the pressure field). However, steady-state two-phase flow in porous media is known to exhibit a regime for which the flow rate depends as a non-linear power law on the global pressure drop, due to the disorder in the capillary barriers at pore throats. A similar nonlinear growth regime was also evidenced experimentally for viscously-unstable drainage in two-dimensional porous media 20 years ago [3]. Here we revisit this flow regime using dynamic pore-network modeling, and explore the non-linearity in the growth properties. We characterize the previously-unstudied dependencies of the statistical finger width and nonlinear growth law's exponent on Ca, and discuss quantitatively, based on theoretical arguments, how disorder in the capillary barriers controls the growth process' non-linearity, and why the flow regime crosses over to Laplacian growth at sufficiently high Ca. In addition, the statistical properties of the fingering patterns are compared to those of Saffman-Taylor fingers, DLA growth patterns, and the results from the aforementioned previous experimental study.