arXiv subjects

Santanu Sinha

Publications and source records attributed to Santanu Sinha.

At least 19 recordsLinked to original sources

arXiv2026

Coordination-number dependent universality in Mixed Wet Percolation

Mixed-wet percolation was introduced recently in the context of two-phase flow in porous media. In this model, the sites of the primal lattice are occupied with a certain probability $p$, and bonds are placed on the dual lattice between two adjacent occupied and unoccupied sites of the primal lattice. The occupied bonds on the dual lattice form perimeter clusters. In this paper, we investigate the scaling properties of the geometric quantities associated with the perimeter clusters of mixed-wet percolation on the dual triangular and dual honeycomb lattices. Although mixed-wet percolation on the dual triangular lattice with a higher coordination number ($z=6$) exhibits ordinary site percolation, the model on the dual honeycomb lattice with a lower coordination number ($z=3$) exhibits the properties of the hull of ordinary site percolation clusters. Such a $z$ dependent breakdown of universality in mixed-wet percolation is rare in the percolation literature. The perimeter clusters in the triangular lattice represent the boundary of the site clusters in the primal lattice, whereas the perimeters in the honeycomb lattice represent their hulls. Because of the low $z$ of the honeycomb lattice, the external and internal perimeters remain isolated. However, the combined external and internal perimeters form cluster boundaries of the site clusters that belong to the site percolation universality class.

cond-mat.stat-mech↗

arXiv2026

Immiscible two-phase flow in porous media: a statistical mechanics approach

The central problem in the physics of immiscible two-phase flow in porous media is to find a proper description of the flow at scales large enough so that the medium may be regarded as a continuum: the scale-up problem. So far, the only workable approach to the multiphase flow scale-up problem has been a set of phenomenological equations that have obvious weaknesses. Attempts at going beyond this relative permeability theory have so far not led to practical applications due to exploding complexity. Edwin T. Jaynes proposed in the fifties a generalization of statistical mechanics to non-thermal systems based on the information theoretical entropy of Shannon. This approach is used to construct a description of immiscible two-phase flow in porous media at the continuum scales, which is directly related to the physics at the pore scale, and at a level of complexity that is manageable. The approach leads to a thermodynamics-like formalism at the continuum scale with all the relations between variables that "normal" thermodynamics has to offer. New emergent variables appear. Among these, the co-moving velocity stands out as a key variable with implications for ordinary thermodynamics. We present here a short review of this approach.

physics.flu-dyn↗

arXiv2026

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

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.

physics.flu-dyn↗

arXiv2025

Pressure drop-flow rate nonlinearity in bubble trains through a capillary bundle

We investigate the effective rheology of a train of elongated bubbles of negligible viscosity flowing in capillary tubes. Building upon the classical Bretherton theory for a single bubble, we extend the analysis to a train of bubbles in a single capillary tube and finally to an array of parallel, noninteracting capillary tubes, i.e., a capillary bundle. Our goal is to characterize the nonlinear pressure drop-flow rate relation of this simplified two-phase system by incorporating the thin-film hydrodynamics at small capillary numbers. We model the structural heterogeneity of the bundle by assuming that the tube radii follow a truncated power-law distribution and examine deviations of the system from the Darcy law in terms of both its statistical properties and the parameters characterizing the bubble train (i.e., the tube slenderness ratio, the volume fraction, and the number of bubbles). The main result is that two-phase flow alters the effective rheology, leading to deviations from Darcy-type behavior across the entire parameter space investigated. Specifically, for a limited number of bubbles, the flow exhibits a smooth transition from the Bretherton regime, where the pressure drop scales with the flow rate to the power of 2/3, to weaker sublinear regimes with exponents between 2/3 and unity. Interestingly, increasing the number of bubbles or narrowing the pore-size distribution leads to only minor deviations from the Bretherton regime. The resulting pressure drop-flow rate exponents are qualitatively similar to those reported in the literature for immiscible two-phase flow in porous media, despite the inherent simplicity of the capillary bundle model.

physics.flu-dyn↗

arXiv2025

Mixed-wet percolation on a dual square lattice

We present a percolation model that is inspired by recent works on immiscible two-phase flow in a mixed-wet porous medium made of a mixture of grains with two different wettability properties. The percolation model is constructed on a dual lattice where the sites on the primal lattice represent the grains of the porous medium, and the bonds on the dual lattice represent the pores in between the grains. The bonds on the dual lattice are occupied based on the two adjacent sites on the primal lattice, which represent the pores where the capillary forces average to zero. The spanning cluster of the bonds, therefore, represents the flow network through which the two immiscible fluids can flow without facing any capillary barrier. It turns out to be a percolation transition of the perimeters of a site percolation problem. We study the geometrical properties at the criticality of the perimeter system numerically. A scaling theory is developed for these properties, and their scaling relations with the respective density parameters are studied. We also verified their finite-size scaling relations. Though the site clusters and their perimeters look very different compared to ordinary percolation, the singular behaviour of the associated geometrical properties remains unchanged. The critical exponents are found to be those of the ordinary percolation.

cond-mat.stat-mech↗

arXiv2025

Thermodynamics-Like Formalism for Immiscible and Incompressible Two-Phase Flow in Porous Media

It is possible to formulate immiscible and incompressible two-phase flow in porous media in a mathematical framework resembling thermodynamics based on the Jaynes generalization of statistical mechanics. We review this approach and discuss the meaning of the emergent variables that appear, agiture, flow derivative and flow pressure, which are conjugate to the configurational entropy, the saturation and the porosity respectively. We conjecture that the agiture, the temperature-like variable, is directly related to the pressure gradient. This has as a consequence that the configurational entropy, a measure of how the fluids are distributed within the porous media and the accompanying velocity field, and the differential mobility of the fluids are related. We also develop elements of another version of the thermodynamics-like formalism where fractional flow rather than saturation is the control variable, since this is typically the natural control variable in experiments.

physics.flu-dyn↗

arXiv2023

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.

physics.flu-dyn↗

arXiv2023

Immiscible two-phase flow in porous media: Effective rheology in the continuum limit

It is becoming increasingly clear that there is a regime in immiscible two-phase flow in porous media where the flow rate depends of the pressure drop as a power law with exponent different than one. This occurs when the capillary forces and viscous forces both influence the flow. At higher flow rates, where the viscous forces dominate, the flow rate depends linearly on the pressure drop. The question we pose here is what happens to the linear regime when the system size is increased. Based on analytical calculations using the capillary fiber bundle model and on numerical simulations using a dynamical network model, we find that the non-linear regime moves towards smaller and smaller pressure gradients as the system size grows.

physics.flu-dyn↗

arXiv2023

Transition from viscous fingers to foam during drainage in heterogeneous porous media

We investigate the behavior of drainage displacements in heterogeneous porous media finding a transition from viscous fingering to foam-like region. A pore network model incorporating the formation of blobs is adopted to study this phenomenon. By imposing a pressure difference between the inlet and outlet, we observe that the displacement pattern undergoes a significant transition from a continuous front of growing viscous fingers to the emergence of foam, which develops and propagates until breakthrough. This transition occurs at a specific distance from the inlet, which we measure and analyze as a function of the viscosity ratio and the capillary number, demonstrating that it follows a non-trivial power-law decay with both the parameters. Moreover, we discuss the relationship between the evolution of the total flow rate and the local pressure drop, showing that the foam developed reduces global mobility. We observe that foam is formed from the fragmentation of viscous fingers beneath the front, and this instability mechanism is connected with fluctuations of the local flow rate, which we analyze both in the viscous fingering region and in the foam region.

cond-mat.soft↗

arXiv2023

Effective rheology of immiscible two-phase flow in porous media consisting of random mixtures of grains having two types of wetting properties

We consider the effective rheology of immiscible two-phase flow in porous media with random mixtures of two types of grains with different wetting properties using a dynamic pore network model under steady-state. Two immiscible fluids A and B flow through the pores between these two types of grains denoted "+" and "-". Fluid A is fully wetting and B is fully non-wetting with respect to "+" grains and opposite with "-" grains. The direction of the capillary forces in the links between two "+" grains is therefore opposite compared to that between two "-" grains, whereas the capillary forces in the links between two opposite types of grains average to zero. For a window of grain occupation probabilities, a percolating regime appears where there is a high probability of having connected paths with zero capillary forces. Due to these paths, no minimum threshold pressure is required to start a flow in this regime. While varying the pressure drop across the porous medium from low to high in this regime, the relation between the volumetric flow rate and the pressure drop goes from being linear to a power law with exponent 2.56 to linear again. Outside the percolation regime, there is a threshold pressure. No linear regime is observed for low pressure drops. When the pressure drop is high enough for there to be flow, we find that the flow rate depends on the excess pressure drop to an exponents around 2.2-2.3. At even higher pressure drops, the relation is linear. We see no change in exponent for the intermediate regime at the percolation critical points where the zero-capillary force paths disappear. We measure the mobility at the percolation threshold at low pressure drops so that the flow rate versus pressure drop is linear. Assuming a power law, the mobility is proportional to the difference between the occupation probability and the critical occupation probability to a power of around 5.7.

physics.flu-dyn↗

arXiv2023

Local Statistics of Immiscible and Incompressible Two-Phase Flow in Porous Media

We consider immiscible and incompressible two-phase flow in porous media under steady-state conditions using a dynamic pore network model. We focus on the fluctuations in a Representative Elementary Area (REA), with the aim to demonstrate that the statistical distributions of the volumetric flow rate and the saturation within the REA become independent of the size of the entire model when the model is large enough. This independence is a necessary condition for developing a local statistical theory for the flow, which in turn opens for the possibility to formulate a description at scales large enough for the typical pore size to be negligible using differential equations.

physics.flu-dyn↗

arXiv2022

Steady-state two-phase flow of compressible and incompressible fluids in a capillary tube of varying radius

We study immiscible two-phase flow of a compressible and an incompressible fluid inside a capillary tube of varying radius under steady-state conditions. The incompressible fluid is Newtonian and the compressible fluid is an inviscid ideal gas. The surface tension associated with the interfaces between the two fluids introduces capillary forces that vary along the tube due to the variation in the tube radius. The interplay between effects due to the capillary forces and the compressibility results in a set of properties that are different from incompressible two-phase flow. As the fluids move towards the outlet, the bubbles of the compressible fluid grow in volume due to the decrease in pressure. The volumetric growth of the compressible bubbles makes the volumetric flow rate at the outlet higher than at the inlet. The growth is not only a function of the pressure drop across the tube, but also of the ambient pressure. Furthermore, the capillary forces create an effective threshold below which there is no flow. Above the threshold, the system shows a weak non-linearity between the flow rates and the effective pressure drop, where the non-linearity also depends on the absolute pressures across the tube.

physics.flu-dyn↗

arXiv2022

A statistical mechanics for immiscible and incompressible two-phase flow in porous media

We construct a statistical mechanics for immiscible and incompressible two-phase flow in porous media under local steady-state conditions based on the Jaynes maximum entropy principle. A cluster entropy is assigned to our lack of knowledge of, and control over, the fluid and flow configurations in the pore space. As a consequence, two new variables describing the flow emerge: The agiture, that describes the level of agitation of the two fluids, and the flow derivative which is conjugate to the saturation. Agiture and flow derivative are the analogs of temperature and chemical potential in standard (thermal) statistical mechanics. The associated thermodynamics-like formalism reveals a number of hitherto unknown relations between the variables that describe the flow, including fluctuations. The formalism opens for new approaches to characterize porous media with respect to multi-phase flow for practical applications, replacing the simplistic relative permeability theory while still keeping the number of variables tractable.

physics.flu-dyn↗

arXiv2022

The Co-Moving Velocity in Immiscible Two-Phase Flow in Porous Media

We present a continuum (i.e., an effective) description of immiscible two-phase flow in porous media characterized by two fields, the pressure and the saturation. Gradients in these two fields are the driving forces that move the immiscible fluids around. The fluids are characterized by two seepage velocity fields, one for each fluid. Following Hansen et al.\ (Transport in Porous Media, 125, 565 (2018)), we construct a two-way transformation between the velocity couple consisting of the seepage velocity of each fluid, to a velocity couple consisting of the average seepage velocity of both fluids and a new velocity parameter, the co-moving velocity. The co-moving velocity is related but not equal to velocity difference between the two immiscible fluids. The two-way mapping, the mass conservation equation and the constitutive equations for the average seepage velocity and the co-moving velocity form a closed set of equations that determine the flow. There is growing experimental, computational and theoretical evidence that constitutive equation for the average seepage velocity has the form of a power law in the pressure gradient over a wide range of capillary numbers. Through the transformation between the two velocity couples, this constitutive equation may be taken directly into account in the equations describing the flow of each fluid. This is e.g., not possible using relative permeability theory. By reverse engineering relative permeability data from the literature, we construct the constitutive equation for the co-moving velocity. We also calculate the co-moving constitutive equation using a dynamic pore network model over a wide range of parameters, from where the flow is viscosity dominated to where the capillary and viscous forces compete.

physics.flu-dyn↗

arXiv2021

Rheology of immiscible two-phase flow in mixed wet porous media: Dynamic pore network model and capillary fiber bundle model results

Immiscible two-phase flow in porous media with mixed wet conditions was examined using a capillary fiber bundle model, which is analytically solvable, and a dynamic pore network model. The mixed wettability was implemented in the models by allowing each tube or link to have a different wetting angle chosen randomly from a given distribution. Both models showed that mixed wettability can have significant influence on the rheology in terms of the dependence of the global volumetric flow rate on the global pressure drop. In the capillary fiber bundle model, for small pressure drops when only a small fraction of the tubes were open, it was found that the volumetric flow rate depended on the excess pressure drop as a power law with an exponent equal to 3/2 or 2 depending on the minimum pressure drop necessary for flow. When all the tubes were open due to a high pressure drop, the volumetric flow rate depended linearly on the pressure drop, independent of the wettability. In the transition region in between where most of the tubes opened, the volumetric flow depended more sensitively on the wetting angle distribution function and was in general not a simple power law. The dynamic pore network model results also showed a linear dependence of the flow rate on the pressure drop when the pressure drop is large. However, out of this limit the dynamic pore network model demonstrated a more complicated behaviour that depended on the mixed wettability condition and the saturation. In particular, the exponent relating volumetric flow rate to the excess pressure drop could take on values anywhere between 1.0 and 1.8. The values of the exponent were highest for saturations approaching 0.5, also, the exponent generally increased when the difference in wettability of the two fluids were larger and when this difference was present for a larger fraction of the porous network.

physics.flu-dyn↗

arXiv2021

Role of pore-size distribution on effective rheology of two-phase flow in porous media

The flow of immiscible fluids inside a porous medium shows non-linearity in the form of a power law in the rheological properties of the fluids under steady state flow conditions. However, different experimental and numerical studies have reported different values for the exponent related to this power law. Here we explore how the rheological properties of the two-phase flow in porous media depends on the distribution of the pore sizes and how it affects the power-law exponent. The pore-size distribution controls fluctuation in the pore radii and their density in a porous material. We present two approaches, analytical calculations using a capillary bundle model and numerical simulations using dynamic pore-network modeling. We observe crossover from a non-linear to linear rheology when increasing the flow rate where the non-linear part is highly affected by the pore-size distribution. We have also carried out the study for different saturations of the two fluids.

physics.flu-dyn↗

arXiv2020

Crack Localization and the Interplay between Stress Enhancement and Thermal Noise

We study the competition between thermal fluctuations and stress enhancement in the failure process of a disordered system by using a local load sharing fiber bundle model. The thermal noise is introduced by defining a failure probability that constitutes the temperature and elastic energy of the fibers. We observe that at a finite temperature and low disorder strength, the failure process, which nucleate in the absence of any thermal fluctuation, becomes spatially uncorrelated when the applied stress is sufficiently low. The dynamics of the model in this limit lies closely to the universality class of ordinary percolation. When applied stress is increased beyond a threshold value, localized fractures appear in the system that grow with time. We identify the boundary between the localized and random failure process in the space of temperature and applied stress, and find that the threshold of stress corresponding to the onset of localized crack growth increases with the increase of temperature.

cond-mat.dis-nn↗

arXiv2020

Phase transitions and correlations in fracture processes where disorder and stress compete

We study the effect of the competition between disorder and stress enhancement in fracture processes using the local load sharing fiber bundle model, a model that hovers on the border between analytical tractability and numerical accessibility. We implement a disorder distribution with one adjustable parameter. The model undergoes a localization transition as a function of this parameter. We identify an order parameter for this transition and find that the system is in the localized phase over a finite range of values of the parameter bounded by a transition to the non-localized phase on both sides. The transition is first order at the lower transition and second order at the upper transition. The critical exponents characterizing the second order transition are close to those characterizing the percolation transition. We determine the spatiotemporal correlation function in the localized phase. It is characterized by two power laws as in invasion percolation. We find exponents that are consistent with the values found in that problem.

cond-mat.dis-nn↗