Dynamics of invasion percolation

Liquid Repellence of Phobic Fiber Networks

The wetting behavior of fiber networks, which are central to many research and industrial applications, can be difficult to predict accurately owing to their complex, heterogeneous structure. The cylindrical pore model, widely used to interpret and predict the forced wetting of hydrophobic porous materials, often does not yield correct results when working with fibrous networks like paper substrates and non-woven fabrics. This is because these materials exhibit variation in pore size, fiber length, and fiber diameter, as well as a reentrant pore geometry. Quantitative prediction of the critical wetting resistance of hydrophobized papers to arbitrary entrant liquids requires a more sophisticated analytical approach that considers this unique fibrous structure and the effect of stochastic variations within the pore matrix. In this work, we directly measure the critical breakthrough pressure for different porous substrates across various wetting entrant liquids. To isolate the effects of the structure and stochastics on critical wetting behavior of fibrous networks, we analyze additional materials strategically chosen for their subsets of structural features. Ultimately, we formulate a method that demonstrates physical reasonableness, numerical accuracy, and the ability to elucidate the effects of pore size, pore size distribution, fiber diameter, fiber diameter distribution, surface wettability, and liquid surface tension on critical breakthrough pressure of liquids through hydrophobic fibrous networks.

Invasion percolation in short-range and long-range disorder background

In the original invasion percolation model, a random number quantifies the role of necks, or generally the quality of pores, ignoring the structure of pores and impermeable regions (to which the invader cannot enter). In this paper, we investigate invasion percolation (IP), taking into account the impermeable regions, the configuration of which is modeled by ordinary and Ising-correlated site percolation (with short-range interactions, SRI), on top of which the IP dynamics is defined. We model the long-ranged correlations of pores by a random Coulomb potential (RCP). By examining various dynamical observables, we suggest that the critical exponents of Ising-correlated cases change considerably only in the vicinity of the critical point (critical temperature), while for the ordinary percolation case the exponents are robust against the occupancy parameter p. The properties of the model for the long-range interactions [LRI (RCP)] are completely different from the normal IP. In particular, the fractal dimension of the external frontier of the largest hole is nearly 4/3 for SRI far from the critical points, which is compatible with normal IP, while it converges to 1.099±0.04 for RCP. For the latter case, the time dependence of our observables is divided into three parts: the power law (short time), the logarithmic (moderate time), and the linear (long time) regimes. The second crossover time is shown to go to infinity in the thermodynamic limit, whereas the first crossover time is nearly unchanged, signaling the dominance of the logarithmic regime. The average gyration radius of the growing clusters, the length of their external perimeter, and the corresponding roughness are shown to be nearly constant for the long-time regime in the thermodynamic limit.

Universality class for loopless invasion percolation models and a percolation avalanche burst model for hydraulic fracturing

Invasion percolation is a model that was originally proposed to describe growing networks of fractures. Here we describe a loopless algorithm on random lattices, coupled with an avalanche-based model for bursts. The model reproduces the characteristic b-value seismicity and spatial distribution of bursts consistent with earthquakes resulting from hydraulic fracturing ("fracking"). We test models for both site invasion percolation and bond invasion percolation. These have differences on the scale of site and bond lengths l. But since the networks are characterized by their large-scale behavior, l≪L, we find small differences between scaling exponents. Though data may not differentiate between models, our results suggest that both models belong to different universality classes.

Pore-Scale Transport and Two-Phase Fluid Structures in Fibrous Porous Layers: Application to Fuel Cells and Beyond

We present pore-scale simulations of two-phase flows in a reconstructed fibrous porous layer. The three-dimensional microstructure of the material, a fuel cell gas diffusion layer, is acquired via X-ray computed tomography and used as input for lattice Boltzmann simulations. We perform a quantitative analysis of the multiphase pore-scale dynamics, and we identify the dominant fluid structures governing mass transport. The results show the existence of three different regimes of transport: a fast inertial dynamics at short times, characterised by a compact uniform front, a viscous-capillary regime at intermediate times, where liquid is transported along a gradually increasing number of preferential flow paths of the size of one–two pores, and a third regime at longer times, where liquid, after having reached the outlet, is exclusively flowing along such flow paths and the two-phase fluid structures are stabilised. We observe that the fibrous layer presents significant variations in its microscopic morphology, which have an important effect on the pore invasion dynamics, and counteract the stabilising viscous force. Liquid transport is indeed affected by the presence of microstructure-induced capillary pressures acting adversely to the flow, leading to capillary fingering transport mechanism and unstable front displacement, even in the absence of hydrophobic treatments of the porous material. We propose a macroscopic model based on an effective contact angle that mimics the effects of the such a dynamic capillary pressure. Finally, we underline the significance of the results for the optimal design of face masks in an effort to mitigate the current COVID-19 pandemic.

Verification of a Dynamic Scaling for the Pair Correlation Function during the Slow Drainage of a Porous Medium

In this Letter we give experimental grounding for the remarkable observation made by Furuberg et al. [Phys. Rev. Lett. 61, 2117 (1988)PRLTAO0031-900710.1103/PhysRevLett.61.2117] of an unusual dynamic scaling for the pair correlation function N(r,t) during the slow drainage of a porous medium. Those authors use an invasion percolation algorithm to show numerically that the probability of invasion of a pore at a distance r away and after a time t from the invasion of another pore scales as N(r,t)∝r^{-1}f(r^{D}/t), where D is the fractal dimension of the invading cluster and the function f(u)∝u^{1.4}, for u≪1 and f(u)∝u^{-0.6}, for u≫1. Our experimental setup allows us to have full access to the spatiotemporal evolution of the invasion, which is used to directly verify this scaling. Additionally, we connect two important theoretical contributions from the literature to explain the functional dependency of N(r,t) and the scaling exponent for the short-time regime (t≪r^{D}). A new theoretical argument is developed to explain the long-time regime exponent (t≫r^{D}).

Computer Simulations of the Tumor Vasculature: Applications to Interstitial Fluid Flow, Drug Delivery, and Oxygen Supply

Tumor vasculature, the blood vessel network supplying a growing tumor with nutrients such as oxygen or glucose, is in many respects different from the hierarchically organized arterio-venous blood vessel network in normal tissues. Angiogenesis (the formation of new blood vessels), vessel cooption (the integration of existing blood vessels into the tumor vasculature), and vessel regression remodel the healthy vascular network into a tumor-specific vasculature. Integrative models, based on detailed experimental data and physical laws, implement, in silico, the complex interplay of molecular pathways, cell proliferation, migration, and death, tissue microenvironment, mechanical and hydrodynamic forces, and the fine structure of the host tissue vasculature. With the help of computer simulations high-precision information about blood flow patterns, interstitial fluid flow, drug distribution, oxygen and nutrient distribution can be obtained and a plethora of therapeutic protocols can be tested before clinical trials. This chapter provides an overview over the current status of computer simulations of vascular remodeling during tumor growth including interstitial fluid flow, drug delivery, and oxygen supply within the tumor. The model predictions are compared with experimental and clinical data and a number of longstanding physiological paradigms about tumor vasculature and intratumoral solute transport are critically scrutinized.

Visualization of water drying in porous materials by X-ray phase contrast imaging

We present in this study results from X-ray tomographic microscopy with synchrotron radiation performed both in attenuation and phase contrast modes on a limestone sample during two stages of water drying. No contrast agent was used in order to increase the X-ray attenuation by water. We show that only by using the phase contrast mode it is possible to achieve enough water content change resolution to investigate the drying process at the pore-scale. We performed 3D image analysis of the time-differential phase contrast tomogram. We show by the results of such analysis that it is possible to obtain a reliable characterization of the spatial redistribution of water in the resolved pore system in agreement with what expected from the theory of drying in porous media and from measurements performed with other approaches. We thus show the potential of X-ray phase contrast imaging for pore-scale investigations of reactive water transport processes which cannot be imaged by adding a contrast agent for exploiting the standard attenuation contrast imaging mode.

Loopless nontrapping invasion-percolation model for fracking

Recent developments in hydraulic fracturing (fracking) have enabled the recovery of large quantities of natural gas and oil from old, low-permeability shales. These developments include a change from low-volume, high-viscosity fluid injection to high-volume, low-viscosity injection. The injected fluid introduces distributed damage that provides fracture permeability for the extraction of the gas and oil. In order to model this process, we utilize a loopless nontrapping invasion percolation previously introduced to model optimal polymers in a strongly disordered medium and for determining minimum energy spanning trees on a lattice. We performed numerical simulations on a two-dimensional square lattice and find significant differences from other percolation models. Additionally, we find that the growing fracture network satisfies both Horton-Strahler and Tokunaga network statistics. As with other invasion percolation models, our model displays burst dynamics, in which the cluster extends rapidly into a connected region. We introduce an alternative definition of bursts to be a consecutive series of opened bonds whose strengths are all below a specified value. Using this definition of bursts, we find good agreement with a power-law frequency-area distribution. These results are generally consistent with the observed distribution of microseismicity observed during a high-volume frack.

Visualization and quantification of two-phase flow in transparent miniature packed beds

Optical microscopy was used to visualize the flow of two phases [British Petroleum (BP) oil and an aqueous surfactant phase] in confined space, three-dimensional, transparent, natural porous media. The porous media consisted of water-wet cryolite grains packed inside cylindrical, glass microchannels, thus producing microscopic packed beds. Primary drainage of BP oil displacing an aqueous surfactant phase was studied at capillary numbers that varied between 10(-6) and 10(-2). The confinement space had a significant effect on the flow behavior. Phenomena of burst motion and capillary fingering were observed for low capillary numbers due to the domination of capillary forces. It was discovered that breakthrough time and capillary number bear a log-log scale linear relationship, based on which a generalized correlation between oil travel distance x and time t was found empirically.

Physical determinants of vascular network remodeling during tumor growth

The process in which a growing tumor transforms a hierarchically organized arterio-venous blood vessel network into a tumor specific vasculature is analyzed with a theoretical model. The physical determinants of this remodeling involve the morphological and hydrodynamic properties of the initial network, generation of new vessels (sprouting angiogenesis), vessel dilation (circumferential growth), vessel regression, tumor cell proliferation and death, and the interdependence of these processes via spatio-temporal changes of blood flow parameters, oxygen/nutrient supply and growth factor concentration fields. The emerging tumor vasculature is non-hierarchical, compartmentalized into well-characterized zones, displays a complex geometry with necrotic zones and "hot spots" of increased vascular density and blood flow of varying size, and transports drug injections efficiently. Implications for current theoretical views on tumor-induced angiogenesis are discussed.

Flow-correlated dilution of a regular network leads to a percolating network during tumor-induced angiogenesis

We study a simplified stochastic model for the vascularization of a growing tumor, incorporating the formation of new blood vessels at the tumor periphery as well as their regression in the tumor center. The resulting morphology of the tumor vasculature differs drastically from the original one. We demonstrate that the probabilistic vessel collapse has to be correlated with the blood shear force in order to yield percolating network structures. The resulting tumor vasculature displays fractal properties. Fractal dimension, Micro-Vascular Density (MVD), blood flow and shear force have been computed for a wide range of parameters.

Vascular network remodeling via vessel cooption, regression and growth in tumors

The transformation of the regular vasculature in normal tissue into a highly inhomogeneous tumor specific capillary network is described by a theoretical model incorporating tumor growth, vessel cooption, neo-vascularization, vessel collapse and cell death. Compartmentalization of the tumor into several regions differing in vessel density, diameter and in necrosis is observed for a wide range of parameters in agreement with the vessel morphology found in human melanoma. In accord with data for human melanoma the model predicts that microvascular density (MVD), regarded as an important diagnostic tool in cancer treatment, does not necessarily determine the tempo of tumor progression. Instead it is suggested that the MVD of the original tissue as well as the metabolic demand of the individual tumor cell plays the major role in the initial stages of tumor growth.

Plastic response of a 2D Lennard-Jones amorphous solid: detailed analysis of the local rearrangements at very slow strain rate

We analyze in detail the atomistic response of a model amorphous material submitted to plastic shear in the athermal, quasi-static limit. After a linear stress-strain behavior, the system undergoes a noisy plastic flow. We show that the plastic flow is spatially heterogeneous. Two kinds of plastic events occur in the system: quadrupolar localized rearrangements, and shear bands. The analysis of the individual motion of a particle shows also two regimes: a hyper-diffusive regime followed by a diffusive regime, even at zero temperature.

Modeling collective behavior of molecules in nanoscale direct deposition processes

We present a theoretical model describing the collective behavior of molecules in nanoscale direct deposition processes such as dip-pen nanolithography. We show that strong intermolecular interactions combined with nonuniform substrate-molecule interactions can produce various shapes of molecular patterns including fractal-like structures. Computer simulations reveal circular and starlike patterns at low and intermediate densities of preferentially attractive surface sites, respectively. At large density of such surface sites, the molecules form a two-dimensional invasion percolation cluster. Previous experimental results showing anisotropic patterns of various chemical and biological molecules correspond to the starlike regime [P. Manandhar et al., Phys. Rev. Lett. 90, 115505 (2003); J.-H. Lim and C. A. Mirkin, Adv. Mater. (Weinheim, Ger.) 14, 1474 (2002); D. L. Wilson et al., Proc. Natl. Acad. Sci. U.S.A. 98, 13660 (2001); M. Su et al., Appl. Phys. Lett. 84, 4200 (2004); R. McKendry et al., Nano Lett. 2, 713 (2002); H. Zhou et al., Appl. Surf. Sci. 236, 18 (2004); G. Agarwal et al., J. Am. Chem. Soc. 125, 580 (2003)].

Flow correlated percolation during vascular remodeling in growing tumors

A theoretical model based on the molecular interactions between a growing tumor and a dynamically evolving blood vessel network describes the transformation of the regular vasculature in normal tissues into a highly inhomogeneous tumor specific capillary network. The emerging morphology, characterized by the compartmentalization of the tumor into several regions differing in vessel density, diameter, and necrosis, is in accordance with experimental data for human melanoma. Vessel collapse due to a combination of severely reduced blood flow and solid stress exerted by the tumor leads to a correlated percolation process that is driven towards criticality by the mechanism of hydrodynamic vessel stabilization.

Cluster evolution in steady-state two-phase flow in porous media

We report numerical studies of the cluster development of two-phase flow in a steady-state environment of porous media. This is done by including biperiodic boundary conditions in a two-dimensional flow simulator. Initial transients of wetting and nonwetting phases that evolve before steady state has occurred, undergo a crossover where every initial pattern is broken up. For flow dominated by capillary effects with capillary numbers in order of 10(-5), we find that around a critical saturation of nonwetting fluid the nonwetting clusters of size have a power-law distribution ns similar to s(-tau)with the exponent tau=1.92+/-0.04 for large clusters. This is a lower value than the result for ordinary percolation. We also present scaling relation and time evolution of the structure and global pressure.

Large-scale numerical simulations of ultrametric long-range depinning

The depinning of an elastic line interacting with a quenched disorder is studied for long-range interactions, applicable to in-plane crack propagation or wetting. An ultrametric distance is introduced instead of the Euclidean distance, allowing for a drastic reduction of the numerical complexity of the problem. Based on large-scale simulations, two to three orders of magnitude larger than previously considered, we obtain a very precise determination of critical exponents which are shown to be indistinguishable from their Euclidean metric counterparts. Moreover, the scaling functions are shown to be unchanged. The choice of an ultrametric distance thus does not affect the universality class of the depinning transition and opens the way to an analytic real-space renormalization-group approach.

Crossover from capillary fingering to viscous fingering for immiscible unstable flow:Experiment and modeling

Invasion percolation with trapping (IPT) and diffusion-limited aggregation (DLA) are simple fractal models, which are known to describe two-phase flow in porous media at well defined, but unphysical limits of the fluid properties and flow conditions. A decade ago, Fernandez, Rangel, and Rivero predicted a crossover from IPT (capillary fingering) to DLA (viscous fingering) for the injection of a zero-viscosity fluid as the injection velocity was increased from zero. [Phys. Rev. Lett. 67, 2958 (1991)]]. We have performed experiments in which air is injected into a glass micromodel to displace water. These experiments clearly demonstrate this crossover as the injection velocity of the air is increased. Furthermore, simulations, using our standard pore-level model, also support the predicted IPT-to-DLA crossover, as well as the predicted power-law behavior of the characteristic crossover length.

Pore-level modeling of drainage: crossover from invasion percolation fingering to compact flow

A pore-level model of drainage, which has been quantitatively validated, is used to study the effect of increased injection rate (i.e., increased capillary number) upon the flow, with matched-viscosity fluids. For small enough capillary number, the flows from the model correctly reproduce the flows from the invasion percolation with trapping (IPWT) model. As the capillary number is increased, the early-time flows mimic those of the IPWT-model, but then deviate towards compact flow at a characteristic time that decreases as the capillary number increases. That is, the larger the capillary number, the sooner the flow crosses over from IPWT flows towards compact (linear) flows.

Interface dynamics and kinetic roughening in fractals

We consider the dynamics and kinetic roughening of single-valued interfaces in two-dimensional fractal media. Assuming that the local height difference distribution function of the fronts obeys Levý statistics with a well-defined power-law decay exponent, we derive analytic expressions for the local scaling exponents. We also show that the kinetic roughening of the interfaces displays anomalous scaling and multiscaling in the relevant correlation functions. For invasion percolation models, the exponents can be obtained from the fractal geometry of percolation clusters. Our predictions are in excellent agreement with numerical simulations.

Invasion percolation with long-range correlations: first-order phase transition and nonuniversal scaling properties

We present the results of extensive Monte Carlo simulations of the invasion percolation model with trapping (TIP) with long-range correlations, a problem which is relevant to multiphase flow in field-scale porous media, such as oil reservoirs and groundwater aquifers, as well as flow in rock fractures. The correlations are generated by a fractional Brownian motion characterized by a Hurst exponent H. We employ a highly efficient algorithm for simulating TIP, and a novel method for identifying the backbone of TIP clusters. Both site and bond TIP are studied. Our study indicates that the backbone of bond TIP is loopless and completely different from that of site TIP. We obtain precise estimates for the fractal dimensions of the sample-spanning cluster (SSC), the minimal path, and the backbone of site and bond TIP, and analyze the size distribution of the trapped clusters, in order to identify all the possible universality classes of TIP with long-range correlations. For site TIP with H > 1/2 the SSC and its backbone are compact, indicating a first-order phase transition at the percolation threshold, while the minimal paths are essentially straigth lines. For H < 1/2 the SSC, its backbone, and the minimal paths are all fractal with fractal dimensions that depend on the Hurst exponent H. The fractal dimension of the loopless backbone for bond TIP is much less than that of site TIP for any H.

Competition between correlated buoyancy and uncorrelated capillary effects during drainage

We study drainage in a horizontally oriented rough fracture joint filled with glass beads. The shape and structure of the drained areas is the result of competition between two effects: (1) variations in the capillary thresholds necessary to be overcome in order to drain the pores and (2) the height variations due to the roughness of the fracture joint. These height variations have long range correlations due to the self-affine nature of the fracture. The capillary thresholds are uncorrelated. We tune the relative strength of these two effects by performing experiments in a centrifuge and thus changing the "strength of gravity." As gravity is increased, the structure of the drained areas change from that of invasion percolation to a structure composed of compact blobs linked together by threadlike links. We study both the geometry and the effect of trapping while changing acceleration of gravity from zero to 6g(0). At high centrifugal acceleration we further observe fragmentation, migration and coalescence of bubbles of fluid inside the drained areas.

Dynamic contrast-enhanced MRI and fractal characteristics of percolation clusters in two-dimensional tumor blood perfusion

Dynamic contrast-enhanced magnetic resonance imaging (DE-MRI) of the tumor blood pool is used to study tumor tissue perfusion. The results are then analyzed using percolation models. Percolation cluster geometry is depicted using the wash-in component of MRI contrast signal intensity. Fractal characteristics are determined for each two-dimensional cluster. The invasion percolation model is used to describe the evolution of the tumor perfusion front. Although tumor perfusion can be depicted rigorously only in three dimensions, two-dimensional cases are used to validate the methodology. It is concluded that the blood perfusion in a two-dimensional tumor vessel network has a fractal structure and that the evolution of the perfusion front can be characterized using invasion percolation. For all the cases studied, the front starts to grow from the periphery of the tumor (where the feeding vessel was assumed to lie) and continues to grow toward the center of the tumor, accounting for the well-documented perfused periphery and necrotic core of the tumor tissue.