Invasion Percolation Research Articles

Viscous Fingering and Preferential Flow Paths in Heterogeneous Porous Media

AbstractPreferential flow channeling and viscous fingering are widely observed phenomena in heterogeneous porous media from the pore scale to the core and reservoir scales. The transition from capillary to viscous fingering with increasing injection rate is also a well‐known phenomenon. Based on the previously observed visual similarity of viscous fingers and preferential single‐phase flow paths in 2‐D simulations, we postulate that these single‐ and two‐phase flow patterns should also be interrelated in 3‐D porous media. Capillary fingering is a manifestation of invasion percolation, a process restricted to the subnetwork of pores obtained from the critical path analysis. We investigated single‐phase flow channeling by applying a method similar to critical path analysis to the flow rate field and compared the subnetwork thus determined to the set of invaded pipes in drainage simulations. Our aim is to quantify the relation between preferential flow paths and viscous fingering, its dependence on pore‐scale heterogeneity and pore connectivity, and ultimately to upscale the network observations to the field scale appropriate for the engineering applications involving drainage in heterogeneous media.

Constrained Invasion Percolation Model: Growth via Leath Bursts and the Origin of Seismic b-Value.

We analyze a new model for growing networks, the constrained Leath invasion percolation model. Cluster dynamics are characterized by bursts in space and time. The model quantitatively reproduces the observed frequency-magnitude scaling of earthquakes in the limit that the occupation probability approaches the critical bond percolation probability in d=2. The model may have application to other systems characterized by burst dynamics.

Gas Flow by Invasion Percolation Through the Hydrate Stability Zone

AbstractWe describe gas flow through the hydrate stability zone as a growing gas network with hydrate forming on its edges. The diffusion‐limited model successfully describes the dynamics of hydrate formation observed experimentally. We obtain an expression for the rate of hydrate growth in a region where gas and water coexist. Our modeled rate of hydrate formation is much lower than predicted with a kinetics‐based approach, and it more successfully simulates the hydrate formation rate observed in geologic systems. Current simulation approaches can readily incorporate our approach to describe hydrate formation in geological systems.

Properties of the Vascular Networks in Malignant Tumors.

This work presents an analysis for real and synthetic angiogenic networks using a tomography image that obtains a portrait of a vascular network. After the image conversion into a binary format it is possible to measure various network properties, which includes the average path length, the clustering coefficient, the degree distribution and the fractal dimension. When comparing the observed properties with that produced by the Invasion Percolation algorithm (IPA), we observe that there exist differences between the properties obtained by the real and the synthetic networks produced by the IPA algorithm. Taking into account the former, a new algorithm which models the expansion of an angiogenic network through randomly heuristic rules is proposed. When comparing this new algorithm with the real networks it is observed that now both share some properties. Once creating synthetic networks, we prove the robustness of the network by subjecting the original angiogenic and the synthetic networks to the removal of the most connected nodes, and see to what extent the properties changed. Using this concept of robustness, in a very naive fashion it is possible to launch a hypothetical proposal for a therapeutic treatment based on the robustness of the network.

Long paths in first passage percolation on the complete graph I. Local PWIT dynamics

We study the random geometry of first passage percolation on the complete graph equipped with independent and identically distributed edge weights, continuing the program initiated by Bhamidi and van der Hofstad [9]. We describe our results in terms of a sequence of parameters $(s_n)_{n\geq 1}$ that quantifies the extreme-value behavior of small weights, and that describes different universality classes for first passage percolation on the complete graph. We consider both $n$-independent as well as $n$-dependent edge weights. The simplest example consists of edge weights of the form $E^{s_n}$, where $E$ is an exponential random variable with mean 1. In this paper, we investigate the case where $s_n\rightarrow \infty$, and focus on the local neighborhood of a vertex. We establish that the smallest-weight tree of a vertex locally converges to the invasion percolation cluster on the Poisson weighted infinite tree. In addition, we identify the scaling limit of the weight of the smallest-weight path between two uniform vertices.

Rumor Spreading and Invasion Percolation

We study the models of rumor spreading and invasion bond percolation aimed at the revelation of possible connections between them. Rumor spreading model describes the dissemination of a rumor due to the periodical repetition of sequential phone calls, whereas the invasion bond percolation refers to the spread of liquid in the porous environment. During a round of the rumor spreading, each node performs a call only once, meanwhile transmitting all the information that it knows at the moment of the call to its neighbors. The rumor reaches the receiver node during one round if there is a chain of successive calls between the source of the rumor and that node. The sequence of calls is taken uniformly at random from the set of all possible sequences (permutations of nodes). We compare the propagation of the rumor spreading with the invasion bond percolation in order to put forth necessary improvements of the percolation rules to map one model onto another, and vice versa

Terrain Amplification with Implicit 3D Features

While three-dimensional landforms, such as arches and overhangs, occupy a relatively small proportion of most computer-generated landscapes, they are distinctive and dramatic and have an outsize visual impact. Unfortunately, the dominant heightfield representation of terrain precludes such features, and existing in-memory volumetric structures are too memory intensive to handle larger scenes. In this article, we present a novel memory-optimized paradigm for representing and generating volumetric terrain based on implicit surfaces. We encode feature shapes and terrain geology using construction trees that arrange and combine implicit primitives. The landform primitives themselves are positioned using Poisson sampling, built using open shape grammars guided by stratified erosion and invasion percolation processes, and, finally, queried during polygonization. Users can also interactively author landforms using high-level modeling tools to create or edit the underlying construction trees, with support for iterative cycles of editing and simulation. We demonstrate that our framework is capable of importing existing large-scale heightfield terrains and amplifying them with such diverse structures as slot canyons, sea arches, stratified cliffs, fields of hoodoos, and complex karst cave networks.

Effect of aperture field anisotropy on two-phase flow in rough fractures

The void space geometry of rough fractures is one of most important factors controlling two-phase flow in fractured media. This paper presents a numerical study on the effect of aperture field anisotropy on two-phase flow properties in rough fractures. By using a power spectrum based method, we generate multiple realizations of synthetic rough fractures with aperture fields of different anisotropy factors. Fluid–fluid displacement in these fractures is simulated by a modified invasion percolation model. It is found that the spatiotemporal distribution of fluid phases is strongly influenced by the aperture field anisotropy. On average, both the nonwetting phase saturation and the entrapped wetting phase saturations at breakthrough decrease with increasing anisotropy factor; but the specific interfacial area is larger for a higher anisotropy ratio. Relative permeabilities to both phases in the direction parallel to the displacement increase with the anisotropy factor, indicating a reduced phase interference due to aperture field anisotropy. Empirical equations are proposed to link the relative permeabilities to the anisotropy factor. These results improve our understanding of immiscible displacement in rough fractures and can be useful for predicting two-phase flow in fractured media at the large scale where geomechanical and chemical processes produce anisotropic roughness.

Impact of Confining Stress on Capillary Pressure Behavior During Drainage Through Rough Fractures

AbstractWe study, numerically, the behavior of capillary pressure (Pc) during slow immiscible displacement in a rough fracture as a function of the degree of fracture aperture heterogeneity that results from two distinct mechanisms: normal confining stress and fracture surface correlation. We generate synthetic self‐affine rough fractures at different correlation scales, solve the elastic contact problem to model the effect of confining stress, and simulate slow immiscible displacement of a wetting fluid by a nonwetting one using a modified invasion percolation model that accounts for in‐plane curvature of the fluid‐fluid interface. Our modeling results indicate that the power spectral density, S(f), of Pc, can be used to qualitatively characterize fracture aperture heterogeneity. We show that the distribution of forward avalanche sizes follows a power law , with exponent α=2, in agreement with previously reported values for porous media and equal to the expected theoretical exponent for a self‐organized criticality process.

Evaluation of van Genuchten-Mualem model on the relative permeability for unsaturated flow in aperture-based fractures

The relative permeability versus saturation relation is fundamental for modeling unsaturated flow processes in aperture-based fractures. On the basis of the local cubic law and the connectivity probability of aperture distribution within single fractures, an analytical relationship between the relative permeability and saturation is established. When the van Genuchten function on the description of the capillary pressure-saturation relationship is applied, an explicit mathematical expression between the relative permeability and saturation is gained similar to the form of the van Genuchten-Mualem (VG-M) model of porous media. Through the comparisons between the several series of laboratory-measured data available in the literature and theoretical results, good agreements are both presented and they suggest that the proposed VG-M model can be successfully used for aperture-based fractures as well as porous media. To understand aperture variation dependence of the relative permeability properties, a computational method combining fractal theory and the invasion percolation model to determine the empirical parameters associated with the spatial correlation of aperture distribution is developed. Simulated fractures with spatially correlated and uncorrelated aperture distributions corresponding respectively to small and large fractal dimensions are both studied. According to the simulated results, the residual saturation generally increases with fractal dimension increments. However, the value of m exhibits a weak dependence on fractal dimension. The proposed model can be applied to various engineering such as seepage control for rock slopes, petroleum reservoirs, nuclear waste disposal and geologic storage of carbon dioxide.

A semi-continuum model of saturation overshoot in one dimensional unsaturated porous media flow

A semi-continuum model for fluid flow in saturated-unsaturated porous medium in one spatial dimension is presented. The model is based on well-established physics, measurable parameters and material characteristics. The porous material is characterized by porosity, intrinsic permeability, main wetting and draining branches of the retention curve, and the saturation dependence of the relative permeability. The fluid is characterized by its density and dynamic viscosity. The only physics involved is the mass balance of fluid in porous media together with the Darcy-Buckingham Law for fluid flow in unsaturated porous media. The model is a cellular automaton based on the Macro Modified Invasion Percolation concept of dividing the porous medium into blocks which are not infinitesimal and are assumed to retain the characteristics of a porous medium. The cellular automaton repeats three successive rules: saturation update in each block, pressure update in each block, and flux update between neighboring blocks. The model tracks the evolution of the relative saturation, the fluid capillary pressure, and the fluid flux. The model is shown to reproduce qualitatively and quantitatively all features of one dimensional saturation overshoot behavior reported in the literature.

Electro‐Thermal Subsurface Gas Generation and Transport: Model Validation and Implications

AbstractGas generation and flow in soil is relevant to applications such as the fate of leaking geologically sequestered carbon dioxide, natural releases of methane from peat and marine sediments, and numerous electro‐thermal remediation technologies for contaminated sites, such as electrical resistance heating. While traditional multiphase flow models generally perform poorly in describing unstable gas flow phenomena in soil, Macroscopic Invasion Percolation (MIP) models can reproduce key features of its behavior. When coupled with continuum heat and mass transport models, MIP has the potential to simulate complex subsurface scenarios. However, coupled MIP‐continuum models have not yet been validated against experimental data and lack key mechanisms required for electro‐thermal scenarios. Therefore, the purpose of this study was to (a) incorporate mechanisms required for steam generation and flow into an existing MIP‐continuum model (ET‐MIP), (b) validate ET‐MIP against an experimental lab‐scale electrical resistance heating study, and (c) investigate the sensitivity of water boiling and gas (steam) transport to key parameters. Water boiling plateaus (i.e., latent heat), heat recirculation within steam clusters, and steam collapse (i.e., condensation) mechanisms were added to ET‐MIP. ET‐MIP closely matched observed transient gas saturation distributions, measurements of electrical current, and temperature distributions. Heat recirculation and cluster collapse were identified as the key mechanisms required to describe gas flow dynamics using a MIP algorithm. Sensitivity analysis revealed that gas generation rates and transport distances, particularly through regions of cold water, are sensitive to the presence of dissolved gases.

Pore network modeling of liquid water and oxygen transport through the porosity-graded bilayer gas diffusion layer of polymer electrolyte membrane fuel cells

In polymer electrolyte membrane fuel cells (PEMFCs), understanding the effect of graded porosity on liquid water distribution and oxygen diffusion inside the porous components is essential for the improvement of water management strategy. In this work, a regular three-dimensional pore network model is constructed to represent the bilayer gas diffusion layer (GDL) consisting of a porosity-graded micro-porous layer (GMPL) and a gas diffusion backing layer (GDBL). Based on this model, the liquid water distribution at breakthrough, relative oxygen effective diffusivity and limiting current density as a function of water saturation are obtained using invasion percolation algorithm and resistor-network theory, respectively. Parametric studies are also conducted to elucidate the effects of several structural factors, such as the graded porosity value of the GMPL, the GMPL thickness and the number of sub-layers in GMPL. The simulated results illustrate that the introduction of graded porosity can relieve the extent of the flooding phenomenon to a certain extent, and the underlying mechanism can be explained from the perspective of the pore scale. Furthermore, an optimum graded porosity value where the material has the preferred oxygen transfer capacity and high limit current density under partially saturated condition (water saturation is between 0.025 and 0.5) can be found.

On Condensation and Evaporation Mechanisms in Disordered Porous Materials.

Sorption isotherm measurement is a standard method for characterizing porous materials. However, such isotherms are generally hysteretic, differing between condensation and evaporation. Quantitative measurement of pore diameter distributions requires proper identification of the mechanisms at play, a topic which has been and remains the subject of intensive studies. In this paper, we compare high-precision measurements of condensation and evaporation of helium in Vycor, a prototypical disordered porous glass, to a model incorporating mechanisms on the single pore level through a semimacroscopic description and collective effects through lattice simulations. Our experiment determines both the average of the fluid density through volumetric measurements and its spatial fluctuations through light scattering. We show that the model consistently accounts for the temperature dependence of the isotherm shape and of the optical signal over a wide temperature range as well as for the existence of thermally activated relaxation effects. This demonstrates that the evaporation mechanism evolves from pure invasion percolation from the sample's surfaces at the lowest temperature to percolation from bulk cavitated sites at larger temperatures. The model also shows that the experimental lack of optical signals during condensation does not imply that condensation is unaffected by network effects. In fact, these effects are strong enough to make most pores to fill at their equilibrium pressure, a situation deeply contrasting the behavior for isolated pores. This implies that, for disordered porous materials, the classical Barrett-Joyner-Halenda approach, when applied to the condensation branch using an extended version of the Kelvin equation, should properly measure the true pore diameter distribution. Our experimental results support this conclusion.

Invasion percolation on Galton-Watson trees

We consider invasion percolation on Galton-Watson trees. On almost every Galton-Watson tree, the invasion cluster almost surely contains only one infinite path. This means that for almost every Galton-Watson tree, invasion percolation induces a probability measure on infinite paths from the root. We show that under certain conditions of the progeny distribution, this measure is absolutely continuous with respect to the limit uniform measure. This confirms that invasion percolation, an efficient self-tuning algorithm, may be used to sample approximately from the limit uniform distribution. Additionally, we analyze the forward maximal weights along the backbone of the invasion cluster and prove a limit law for the process.

Pore Network Investigation of Trapped Gas and Foam Generation Mechanisms

The mobility of gas is greatly reduced when the injected gas is foamed. The reduction in gas mobility is attributed to the reduction in gas relative permeability and the increase in gas effective viscosity. The reduction in the gas relative permeability is a consequence of the larger amount of gas trapped when foam is present while the increase in gas effective viscosity is explicitly a function of foam texture. Therefore, understanding how foam is generated and subsequent trapped foam behavior is of paramount importance to modeling of gas mobility. In this paper, we push the envelope to enlighten our decisions of which descriptions are most physical to foam flow in porous media regarding both the flowing foam fraction and the rate of generation. We use a statistical pore network interwoven with the invasion percolation with memory algorithm to model foam flow as a drainage process and investigate the dependence of the flowing foam fraction on the pressure gradient and to shed light on foam generation mechanisms. A critical snap-off probability is required for strong foam to emerge in our network. The pressure gradient and, hence, the gas mobility reduction are very low below this critical snap-off probability. Above this snap-off probability threshold, we find that the steady-state flowing lamellae fraction scales as $$(\nabla \tilde{p})^{0.19}$$ in 2D lattices and as $$(\nabla \tilde{p})^{0.32}$$ in 3D lattices. Results obtained from our network were convolved with percolation network scaling ideas to compare the probabilities of snap-off and lamella division mechanisms in the network during the initial gas displacement at the leading edge of the gas front. At this front, during strong foam flow, lamella division is practically nonexistent in 2D lattices. In 3D lattices, lamella division occurs, but the probability of snap-off is always greater than the probability of lamella division.

Drying and percolation in correlated porous media

We study how the dynamics of a drying front propagating through a porous medium are affected by small-scale correlations in material properties. For this, we first present drying experiments in micro-fluidic micro-models of porous media. Here, the fluid pressures develop more intermittent dynamics as local correlations are added to the structure of the pore spaces. We also consider this problem numerically, using a model of invasion percolation with trapping, and find that there is a crossover in invasion behaviour associated with the length-scale of the disorder in the system. The critical exponents associated with large enough events are similar to the classic invasion percolation problem, whereas the addition of a finite correlation length significantly affects the exponent values of avalanches and bursts, up to some characteristic size. This implies that the even a weak local structure can interfere with the universality of invasion percolation phenomena.

Pore network modeling of fibrous porous media of uniform and gradient porosity

Two phase flow phenomena in fibrous media is studied using pore network modeling and visualized in 3D space. To explore the effect of network structure on flow property, virtual 3D fibrous mat with uniform and gradient porosity are constructed respectively using the realistic description of the complex fiber structure obtained from X-ray tomographic imaging. A marker-based watershed method is presented to extract pore networks directly from 3D segmented images. Invasion percolation algorithm is then applied to the extracted pore network to simulate primary drainage. The capillary pressure curve generated from drainage simulation is compared with an image-based full morphology method and achieves good agreement. The absolute and relative permeability are performed. For absolute permeability, we detected that the anisotropy of permeability tensor of layered fibrous mat can't be accurately reflected when pore cells are simplified using inscribed maximal balls. For relative permeability curves plotted as functions of water saturation, the results indicate a significant impact of the flow direction on liquid water distribution for porous material with gradient porosity.

The scaling limits of the Minimal Spanning Tree and Invasion Percolation in the plane

We prove that the Minimal Spanning Tree and the Invasion Percolation Tree on a version of the triangular lattice in the complex plane have unique scaling limits, which are invariant under rotations, scalings, and, in the case of the $\mathsf{MST}$, also under translations. However, they are not expected to be conformally invariant. We also prove some geometric properties of the limiting $\mathsf{MST}$. The topology of convergence is the space of spanning trees introduced by Aizenman et al. [Random Structures Algorithms 15 (1999) 319–365], and the proof relies on the existence and conformal covariance of the scaling limit of the near-critical percolation ensemble, established in our earlier works.

A 3D model of hydraulic fracturing and microseismicity in anisotropic stress fields

We present a 3D numerical model for hydraulic fracturing and damage of low permeable rock in an anisotropic stress field. The model computes the intermittent damage propagation, microseismic event-locations, microseismic event-distribution, damaged rock volume and injection pressure. The model builds on concepts from invasion percolation theory, where cells in a regular grid are connected by transmissibilities, also called bonds. An intact bond breaks when the fluid pressure exceeds the least compressive stress and random uniformly distributed bond strength. Analytical expressions are obtained for the fractions of the weakest bonds in the x-, y- and z-directions as a function of the stress anisotropy and the rock strength anisotropy. A strategy is suggested for a fast solution of the pressure equation when the fluid pressure is restricted to the damaged rock volume. An expression for the volume of damaged rock is obtained in the cases where it has a high permeability. The model is demonstrated on a published case from the Barnett Shale and it reproduces the observed main features, such as the spatial and temporal distribution of the events, the magnitude–frequency distribution and the injection pressure. The microseismic event-distribution and the b-value depend on the permeability of the damaged rock volume. The b-value increases with decreasing permeability from a little < 0.6 to a value above 2 for the maximum possible permeabilities. The damaged rock volume is non-compact and similar to a percolation cluster for “high” damaged rock permeabilities, and it becomes increasingly compact with decreasing permeabilities. The resulting loop-less fracture network is found to have similar characteristics for different damaged rock permeabilities.