PhASE-Flow: Spin-Glass Model for Two-Phase Flow

• PhASE-Flow is a phase-based statistical mechanics framework that maps pore-scale configurations of immiscible two-phase flow onto a disordered Ising spin system.
• It employs a dynamic pore network model and maximum-entropy inference via Boltzmann machine learning to infer effective couplings and build a phase diagram in (Sₙ, Ca) space.
• The model predicts a clear transition from paramagnetic to spin-glass phases, linking microscopic interface organization with macroscopic nonlinear flow behavior and hysteresis.

PhASE-Flow denotes, in the terminology introduced for steady-state immiscible two-phase flow in porous media, a phase-based statistical-mechanics description in which pore-scale fluid configurations are mapped onto a disordered Ising spin system and analyzed through spin-glass theory. In this construction, steady-state flow patterns generated by a dynamic pore network model are converted into spin configurations, an effective spin-glass Hamiltonian is inferred by the Jaynes maximum entropy principle with machine learning, and the resulting model is used to build a phase diagram in $(S_n,\mathrm{Ca})$. The central result is that the critical line separating paramagnetic and spin-glass phases coincides with the crossover at which Darcy-scale transport changes from linear to non-linear, thereby linking pore-scale interface organization, macroscopic flow laws, hysteresis, and broad-band fluctuations within a single framework (Sinha et al., 9 Mar 2026).

1. Darcy-scale problem and phase-based reformulation

At the Darcy scale, immiscible two-phase flow in porous media is described by relations among volumetric flow rate $Q$, pressure drop $\Delta P$, and non-wetting saturation $S_n$. The regime structure considered in PhASE-Flow is the established linear–nonlinear–linear sequence in capillary number: at low and high $\mathrm{Ca}$,

$Q \propto \Delta P,$

whereas at intermediate $\mathrm{Ca}$,

$Q \sim (\Delta P - P_t)^\alpha,\qquad \alpha>1.$

Within the classification attributed to Berg and co-workers, steady-state regimes are separated into Ia, Ib, II, and III: Ia is linear flow with interfaces essentially frozen, Ib is linear flow with strongly fluctuating interfaces and strong hysteresis, II is the nonlinear power-law regime, and III is the high-$\mathrm{Ca}$ linear regime.

PhASE-Flow reformulates the corresponding closure problem as a phase problem. Rather than deriving a Darcy-scale constitutive law directly from pore-scale hydrodynamics, it asks whether the ensemble of steady-state pore configurations can be represented by an effective disordered spin model whose phases predict macroscopic transport transitions. In that sense, the framework treats Darcy-scale flow regimes as emergent phases of an inferred statistical ensemble.

2. Dynamic pore network model and spin representation

The microscopic model is a dynamic pore network on a two-dimensional diamond lattice with $L\times L$ links, with $Q$0 in the main data production. Each link is a composite hourglass-shaped pore of length $Q$1 and random radius $Q$2. Two immiscible fluids, wetting and non-wetting, are assigned equal viscosities, $Q$3, so the viscosity ratio is $Q$4, and the surface tension is $Q$5. Link flow obeys a viscous–capillary balance,

$Q$6

with $Q$7,

$Q$8

and hourglass capillary pressures

$Q$9

Kirchhoff’s law at nodes,

$\Delta P$0

closes the network pressure system. The boundary conditions are periodic in both directions, the system is closed, and the global saturation $\Delta P$1 is fixed. The capillary number is controlled through the total flow,

$\Delta P$2

After transients of approximately $\Delta P$3 pore volumes, configurations are recorded with spacing $\Delta P$4 time steps to avoid temporal correlations. The model excludes film and corner flow and generates steady-state patterns of bubbles and ganglia that qualitatively reproduce regimes Ia, Ib, II, and III (Sinha et al., 9 Mar 2026).

The spin mapping is defined linkwise from the non-wetting saturation $\Delta P$5. For each link,

$\Delta P$6

where $\Delta P$7 is the average pore-scale non-wetting saturation in a configuration or ensemble. This thresholding does not enforce zero magnetization: if non-wetting occupancy is globally biased, the spin ensemble remains biased as well. Repeated time slices across many disorder realizations yield an empirical ensemble of spin configurations.

The corresponding effective Hamiltonian is

$\Delta P$8

Here $\Delta P$9 encode local biases associated with geometry and local saturation, while $S_n$0 are effective pairwise couplings. The quenched disorder of the porous medium is transferred to quenched disorder in $S_n$1. Empirically, low-$S_n$2 coupling distributions are often bimodal, with significant positive and negative modes and marked sample-to-sample variation, whereas at high $S_n$3 the coupling distribution becomes more Gaussian and centered near zero. The former is characteristic of frustration; the latter indicates weaker and more homogeneous effective interactions.

3. Maximum-entropy inference and verification

The inferred spin ensemble is constructed by imposing agreement with one- and two-point statistics. Maximizing the Shannon entropy

$S_n$4

subject to normalization and constraints on $S_n$5 and $S_n$6 yields

$S_n$7

with partition function

$S_n$8

Thus the maximum-entropy model constrained by the observed one- and two-point functions is exactly an Ising-like spin glass.

Parameter estimation is performed by inverse Ising inference, implemented as Boltzmann machine learning. From the pore-network data one computes $S_n$9 and $\mathrm{Ca}$0; for a candidate parameter set one generates Monte Carlo spin samples and computes the corresponding model moments. Learning minimizes the Kullback–Leibler distance between empirical and model distributions, yielding the standard updates

$\mathrm{Ca}$1

$\mathrm{Ca}$2

The loss function is the mean-squared mismatch in one- and two-point statistics, and the learning rate is adapted with a ReduceLROnPlateau scheduler.

To keep Monte Carlo inference tractable, the original $\mathrm{Ca}$3 pore-network configurations are coarse-grained to a $\mathrm{Ca}$4 spin system, so $\mathrm{Ca}$5 and the number of pairwise couplings is $\mathrm{Ca}$6. Monte Carlo sampling uses $\mathrm{Ca}$7 sweeps for equilibration and $\mathrm{Ca}$8 sweeps for averaging at each Boltzmann machine iteration, with at least $\mathrm{Ca}$9 learning iterations per parameter set. Validation is performed at the level of two-point and three-point statistics: two-point correlation matrices from the pore network and from Monte Carlo sampling are visually indistinguishable, scatter plots of $Q \propto \Delta P,$0 versus $Q \propto \Delta P,$1 and $Q \propto \Delta P,$2 versus $Q \propto \Delta P,$3 lie almost on the diagonal, and unconstrained three-point correlations also agree well. This supports the claim that the maximum-entropy spin-glass model captures the key statistics of the non-equilibrium steady-state flow configurations (Sinha et al., 9 Mar 2026).

4. Phase diagram and correspondence with Darcy-scale regimes

With the effective Hamiltonian established, PhASE-Flow analyzes phases through standard spin-glass observables. For each disorder realization and parameter set $Q \propto \Delta P,$4, the site magnetizations are averaged over steady-state configurations, and disorder-averaged order parameters are defined as

$Q \propto \Delta P,$5

Here $Q \propto \Delta P,$6 measures ferromagnetic order, while $Q \propto \Delta P,$7 is the Edwards–Anderson order parameter. The corresponding susceptibilities are

$Q \propto \Delta P,$8

and

$Q \propto \Delta P,$9

The standard interpretations are retained: paramagnetic $\mathrm{Ca}$0 corresponds to $\mathrm{Ca}$1; ferromagnetic $\mathrm{Ca}$2 to $\mathrm{Ca}$3; and spin glass $\mathrm{Ca}$4 to $\mathrm{Ca}$5.

For the representative case $\mathrm{Ca}$6, $\mathrm{Ca}$7 remains small and nearly flat as $\mathrm{Ca}$8 varies, and $\mathrm{Ca}$9 shows no peak, indicating the absence of a ferromagnetic transition. By contrast, $Q \sim (\Delta P - P_t)^\alpha,\qquad \alpha>1.$0 rises sharply as $Q \sim (\Delta P - P_t)^\alpha,\qquad \alpha>1.$1 decreases, and $Q \sim (\Delta P - P_t)^\alpha,\qquad \alpha>1.$2 exhibits a distinct maximum at $Q \sim (\Delta P - P_t)^\alpha,\qquad \alpha>1.$3, which is the signature of a paramagnetic-to-spin-glass transition. The global phase diagram is assembled from 17 saturations $Q \sim (\Delta P - P_t)^\alpha,\qquad \alpha>1.$4 in steps of $Q \sim (\Delta P - P_t)^\alpha,\qquad \alpha>1.$5, 53 capillary numbers in $Q \sim (\Delta P - P_t)^\alpha,\qquad \alpha>1.$6, 100 network samples per parameter set, and 1000 steady-state configurations per sample. Across these 901 parameter sets, $Q \sim (\Delta P - P_t)^\alpha,\qquad \alpha>1.$7 and $Q \sim (\Delta P - P_t)^\alpha,\qquad \alpha>1.$8 show no clear phase boundaries, whereas $Q \sim (\Delta P - P_t)^\alpha,\qquad \alpha>1.$9 and $\mathrm{Ca}$0 cleanly separate a paramagnetic region from a spin-glass region. The critical capillary number decreases as $\mathrm{Ca}$1 approaches approximately $\mathrm{Ca}$2 (Sinha et al., 9 Mar 2026).

The macroscopic flow curves establish the Darcy-scale connection. For intermediate saturations, plots of $\mathrm{Ca}$3 versus $\mathrm{Ca}$4 show three regimes: low-$\mathrm{Ca}$5 regime I with slope approximately $\mathrm{Ca}$6, intermediate regime II with slope greater than $\mathrm{Ca}$7, and high-$\mathrm{Ca}$8 regime III with slope again approximately $\mathrm{Ca}$9. The decisive observation is that the pressure where $L\times L$0 peaks lies at the crossover between the low-$L\times L$1 linear regime and the nonlinear regime for $L\times L$2–$L\times L$3. In the language of Berg’s classification, the spin-glass critical line matches the Ib–II transition line. Outside that saturation interval the dynamic pore network does not cleanly display a distinct I–II crossover, and the match deteriorates.

Spin-model phase	Order-parameter signature	Flow interpretation
Paramagnetic $L\times L$4	$L\times L$5	Regime II and III, and perhaps Ia
Spin glass $L\times L$6	$L\times L$7	Regime Ib: linear response with strong local freezing
Ferromagnetic $L\times L$8	$L\times L$9	No clear phase boundary observed

This correspondence is the defining encyclopedic feature of PhASE-Flow: it recasts the onset of nonlinear Darcy behavior as the disappearance of a microscopic glassy phase.

5. Dynamic glass state, hysteresis, and fluctuations

Within this framework, the low-$Q$00 spin-glass phase is interpreted as a dynamic glass state. At high $Q$01, spin configurations appear random and $Q$02 is near zero over most sites, which is the expected paramagnetic pattern. At low $Q$03, $Q$04 becomes large across most of the system, corresponding to local freezing and large $Q$05; in the representative $Q$06 case the low-$Q$07 state has $Q$08. Simultaneously, there remain pathways where $Q$09 is small, interpreted as persistent flow channels. The resulting picture is one of long-lived ganglion structures, conduction through frozen pathways with minimal reconfiguration, and intermittent interface motion over a broad range of time scales (Sinha et al., 9 Mar 2026).

Hysteresis follows naturally in the same language. Regime Ib is empirically associated with strong hysteresis in both flow curves and microscopic configurations under cycling of $Q$10 or $Q$11, together with large pressure fluctuations and intermittent ganglion mobilization or stranding. In the spin-glass representation, the corresponding steady states occupy multiple metastable configurations separated by barriers in a rugged energy landscape generated by frustrated $Q$12. Macroscopic response then depends on history. The paper does not provide explicit hysteresis loops in the present dataset, but it links the spin-glass phase to prior reports that regime II does not show hysteresis whereas regime Ib does. Within PhASE-Flow, the spin-glass phase is therefore the regime in which hysteresis and multiscale fluctuations are expected.

6. Nonequilibrium status, limitations, and implications

PhASE-Flow rests on a deliberate conceptual tension. The underlying two-phase flow is dissipative and non-equilibrium, but the inferred spin model is formulated with equilibrium statistical mechanics. The justification offered is not that the fluid itself is in thermodynamic equilibrium, but that the steady-state ensemble of configurations can be treated by a maximum-entropy principle. In this view, the Boltzmann weight is an equilibrium-like representation of the statistics of a non-equilibrium steady state, and the temperature in

$Q$13

is an effective quantity related to agitation rather than the physical temperature of the fluids (Sinha et al., 9 Mar 2026).

Several limitations are explicit. The inferred couplings $Q$14 reproduce the measured correlations but depend sensitively on algorithmic choices such as the learning rate, and the physical interpretation of that sensitivity remains unresolved. The relation between the model temperature and the physical balance of capillary and viscous effects is also not fixed. In addition, the mapping is statistical rather than dynamical: it predicts ensemble properties and phase boundaries, but not a transient evolution law.

Even with those limits, the framework has a definite programmatic significance. It provides a Darcy-scale phase diagram from a microscopic model without explicitly closing hydrodynamic equations at the macroscopic scale. Threshold pressures, intermittency, and hysteresis are interpreted as consequences of glassy constraints and frustration at the pore scale. The fractional-flow data show that curves at different $Q$15 collapse around $Q$16 near $Q$17, where both fluids move with similar velocities, capillary effects are minimized, and the spin-glass transition occurs at the lowest capillary number. Proposed extensions include direct inference from experimental image sequences, application to three-dimensional networks, different wettabilities and viscosity ratios, and systems with film or corner flow. The intended application areas include reservoir engineering, hydrology and soil physics, and reactive or multiphysics transport problems in which steady-state two-phase morphology governs macroscopic response.