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.

Figures (11)

• Figure 1: (a) A pore network consisting $16\times 16$ links on a 2D diamond lattice. The bonds in the lattice represent composite pores and the black dots represent the position of nodes. The thickness of the bonds indicates their average diameter. The gray and blue parts of the links indicate wetting and non-wetting bubbles. There is periodic boundary condition in both the horizontal and vertical directions, which is indicated by the green nodes on the vertical boundaries being identical nodes. The same is true for the red nodes on the horizontal boundaries. Shape of one composite link in terms of interfacial pressure ($p_i'$) is shown in (b) bottom where $\epsilon_1$, $\epsilon_2$ and $\epsilon_3$ show the position of three interfaces. The variation of $p'_i$ at an interface along the length of the tube follows Equation (\ref{['eq_pc']}), which is indicated in (b) top.
• Figure 2: Matrix plots of $\langle\sigma_i\sigma_j\rangle$ comparing the pore network simulation data and the MC sampling data after learning a specific network sample. The plots here are for $S_n=0.3$, and we show two different Ca. For both, the maps from MC are visually indistinguishable from the data, indicating the convergence of the Boltzmann machine learning computations.
• Figure 3: Plots of $\langle\sigma_i\rangle^{\rm Data}$ vs $\langle\sigma_i\rangle^{\rm MC}$ (top row), and $\langle\sigma_i\sigma_j\rangle^{\rm Data}$ vs $\langle\sigma_i\sigma_j\rangle^{\rm MC}$ (bottom row), showing the comparison between DPN simulation data and BML computations. We show data for two saturations $S_n=0.3$ and $0.5$, with two capillary numbers, ${\rm Ca} = 1.0\times 10^{-2}$ and $1.0\times 10^{-4}$ for each $S_n$. Each plot contains results for $10$ different network samples.
• Figure 4: Plots of 3-point correlations $\langle\sigma_i\sigma_j\sigma_k\rangle$ compared between the DPN data and MC samples. We show results for two saturations $S_n=0.3$ and $0.5$, and two capillary numbers ${\rm Ca}=1.0\times 10^{-2}$ and $1.0\times 10^{-4}$ for each $S_n$. Each plot contains results for $10$ different network samples, and we plotted data only for $1000$ randomly chosen unique triples {$ijk$} for each sample.
• Figure 5: Histograms of local field constants $h_i$ for $S_n=0.3$ at different capillary numbers. Each plot shows data for $10$ different network samples.