Glassy phase transition in immiscible steady-state two-phase flow in porous media

Abstract: Two-phase flow in porous media is a ubiquitous phenomenon that has been studied for well over a century. However, we still lack a successful theory that predicts flow on a macroscopic length scale (the so-called Darcy scale) on the basis of a "microscopic" model. Here we show that the characteristic features of two-phase flow on the Darcy scale can be predicted by mapping the distribution of droplets in 2-phase flow onto the distribution of spins in a spin-glass model. The success of this approach is surprising, as two-phase flow is a non-equilibrium phenomenon, whereas the properties of the spin glass are obtained using equilibrium statistical mechanics. To obtain this mapping, we follow the approach of Meshulam and Bialek (Rev. Mod. Phys. 97, 045002 (2025)) and use the Jaynes maximum entropy principle to derive the spin-glass Hamiltonian using machine learning trained on many realizations of the two-phase flow pattern in a dynamic pore network model. With this mapping, we can construct a "phase diagram" for the 2-phase flow system. We find that the critical line separating the paramagnetic phase from a spin glass phase coincides with the transition where the dependence of the rate of two-phase flow on the imposed pressure gradient changes from linear to non-linear. The glassy phase of the spin model coincides with a flow regime characterized by hysteresis and strong fluctuations over a wide range of time scales. It is tempting to identify this flow regime as a dynamic glass state.