# Browsing by Subject "Percolation"

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Item A lattice model for gas production from hydrofractured shale(2016-12) Eftekhari, Behzad; Patzek, Tadeusz W.; Marder, Michael P., 1960-; Olson, Jon E; Sepehrnoori, Kamy; Espinoza, David NShow more Natural gas production from US shale and tight oil plays has increased over the past 10 years, currently constitutes more than half of the total US dry natural gas production, and is projected to provide the US with a major energy source in the next several decades. The increase in shale gas production is driven by advances in hydraulic fracturing. Recent studies have shown that gas production from hydraulically fractured shales has to come from a network of connected hydraulic and natural fractures, and that if one takes the shale permeability to be 10 nD, then the characteristic spacing of the fracture network will be about 1.5 − 3 m. The precise nature of the characteristic spacing, as well as other production and formation properties of the fracture network, are questions which motivated the present dissertation. This dissertation studies (1) the topology of the fracture network, (2) the mechanics of how the fracture network evolves in time during injection and (3) how fracture network geometry affects production. We use percolation theory to study fracture network topology. Fracture are placed on the bonds of a two–dimensional square lattice and follow a power law length distribution. We analytically obtain the scaling of connectivity for power law fracture networks, and numerically compute the percolation threshold as a function of the exponent. We develop a hydrofracture model which makes it possible to simulate initiation and propagation of hydraulic fractures, as well as the interaction between hydraulic and natural fractures. The model uses the Reynolds lubrication approximation to describe fluid flow through the fractures and relies on analytical estimates to predict the stress response. We develop a diffusion model to compute gas production from hydraulically fractured shales. The model uses a random walk algorithm and takes the fracture network as the absorbing boundary to the gas transport equation. We show that scaling the cumulative production versus time data from the diffusion model with respect to characteristic scales of production maps the production versus time plots onto a single scaling curve. Using the model, we identify, or define, characteristic spacing for fracture networks.Show more Item Effects of Heterogeneous and Clustered Contact Patterns on Infectious Disease Dynamics(Public Library of Science, 2011-06-02) Volz, Erik M.; Miller, Joel C.; Galvani, Alison; Meyers, Lauren AncelShow more The spread of infectious diseases fundamentally depends on the pattern of contacts between individuals. Although studies of contact networks have shown that heterogeneity in the number of contacts and the duration of contacts can have far-reaching epidemiological consequences, models often assume that contacts are chosen at random and thereby ignore the sociological, temporal and/or spatial clustering of contacts. Here we investigate the simultaneous effects of heterogeneous and clustered contact patterns on epidemic dynamics. To model population structure, we generalize the configuration model which has a tunable degree distribution (number of contacts per node) and level of clustering (number of three cliques). To model epidemic dynamics for this class of random graph, we derive a tractable, low-dimensional system of ordinary differential equations that accounts for the effects of network structure on the course of the epidemic. We find that the interaction between clustering and the degree distribution is complex. Clustering always slows an epidemic, but simultaneously increasing clustering and the variance of the degree distribution can increase final epidemic size. We also show that bond percolation-based approximations can be highly biased if one incorrectly assumes that infectious periods are homogeneous, and the magnitude of this bias increases with the amount of clustering in the network. We apply this approach to model the high clustering of contacts within households, using contact parameters estimated from survey data of social interactions, and we identify conditions under which network models that do not account for household structure will be biased.Show more Item Hydrological connectivity in vegetated river deltas : the effects of spatial variability and patchiness on channel-island exchange(2018-05-04) Wright, Kyle Austin; Passalacqua, PaolaShow more River deltas are threatened regions of great societal and environmental importance, and their continued survival depends upon a greater understanding of their formation and evolution. Hydrological connectivity in river deltas is important for delivering flow and sediment to the island interior and is responsible for a large portion of the ecosystem benefits that deltas provide, which could be leveraged for restoration projects using nature-based engineering. However, the process is still poorly understood. The roughness of island vegetation is known to significantly limit channel-island connectivity, but the importance of the spatial distribution of vegetation is, as-of-yet, unknown. Using a 2D hydrodynamic model, we investigate the influence of vegetation percent cover, patch size, and stem density on the fraction of discharge allocated to the islands of an idealized delta complex, modeled after the Wax Lake Delta in coastal Louisiana. We find that spatial heterogeneity can substantially alter connectivity when vegetation is dense and covers less than a “disconnectivity” threshold near 50% of the island domain, near the theoretical percolation limit. Above this threshold, models can accurately approximate vegetation as uniform. Below this threshold, however, preferential flow-paths develop in the islands, which greatly alter the hydraulics, transport capabilities, and residence time distribution of the delta complex, with respect to what is seen in uniform vegetation cases. Our results suggest that patchiness has substantial hydrogeomorphic and biogeochemical implications which should be considered when modeling deltaic systems.Show more Item On the Poisson Follower Model(2020-08-14) Dragović, Nataša; Baccelli, F. (François), 1954-; De Veciana, Gustavo; Zitkovic, Gordan; Tran, Ngoc; Taillefumier, Thibaud OShow more This dissertation presents studies of dynamics over the Poisson point process. In particular, we study a special case of Hegselmann-Krause Dynamics [1] over ℝ². Chapter 1 is a brief introduction to the thesis and its structure. Chapter 2 introduces the notation, the definitions and examples of phenomena of interest. In Chapter 3, we go deeper in analyzing the phenomena described by calculating frequency of these phenomena. A system of quadratic inequalities will be introduced to allow one to calculate the probabilities of the events pertaining to this dynamics using methods from integral geometry. Chapter 4 uses percolation arguments to prove the absence of percolation at step 1. In Chapter 5, we provide geometric results of independent interest pertaining to the Follower Dynamics. In Chapter 6, we discuss the limiting behavior of this process and include some more simulations. In Chapter 7 we propose future steps and discuss more general Hegselmann-Krause Dynamics.Show more Item Pore fluid percolation and flow in ductile rocks(2016-08) Ghanbarzadeh, Soheil; Prodanović, Maša; Hesse, Marc; Sepehrnoori, Kamy; Ebrom, Daniel; Bryant, Steven L; DiCarlo, DavidShow more Ductile rocks have capacity to deform in response to large strains without macroscopic fracturing. Such behavior may occur in rocks that did not undergo diagenesis, in weak materials such as rock salt or at greater depths in all rock types where higher temperatures promote crystal plasticity and higher confining pressures suppress brittle fracture (partially molten rocks). The pore network topology and fluid distribution in ductile rocks are governed by textural equilibrium. Therefore, textural equilibrium controls the distribution of the liquid phase in many naturally occurring porous materials such as partially molten rocks and alloys, salt-brine and ice-water systems. In this dissertation, we present a level set method to compute an implicit representation of the liquid-solid interface in textural equilibrium with space-filling tessellations of multiple solid grains in three dimensions. In ductile rocks, pore geometry evolves to minimize the solid-liquid interfacial energy while maintaining a constant dihedral angle, θ, at solid-liquid contact lines. Interfacial energy minimization with level set method is achieved by evolving the solid-liquid interface under surface diffusion to constant mean curvature surface. The liquid volume and dihedral angle constraints are added to the formulation using virtual convective and normal velocity terms. This results in a initial value problem for a system of nonlinear coupled PDEs governing the evolution of the level sets for each grain. A domain decomposition scheme is devised to restrict the computational domain of each grain to few grid points around the grain. The coupling between the interfaces is achieved in a higher level on the original computational domain. Our results show that the grain boundaries with the smallest area can be fully wetted by the pore fluid even for θ > 0. This was previously not thought to be possible at textural equilibrium and reconciles the theory with experimental observations. Even small anisotropy in the fabric of the porous medium allows the wetting of these faces at very low porosities, ϕ < 3%. Percolation and orientation of the wetted faces relative to the anisotropy of the fabric are controlled by θ. We have studied the fluid percolation and percolation thresholds in regular and irregular media. The results show that the pore space is connected at any non-zero porosity when θ ≤ 60°, and percolation threshold in an irregular media comprised of grains with different shapes and sizes is much higher than previously thought. Our results show that the pore network connectivity in ductile rocks is affected by the history of the systems and hysteresis determines the percolation when θ > 60°. We have also computed permeability of the pore networks in different porosities and dihedral angles for both regular and irregular media using Lattice Boltzmann method. Furthermore, we studied the effects of grain texture anisotropy on the permeability anisotropy. Until recent years, rock salt has been considered to be impermeable as it seems to contains and keep gas inclusions for long time. Increasing energy demand and necessity of producing hydrocarbon reservoir enclosed or touched by salt deposit and also urgent need of safe repository sites for high-level nuclear waste have brought attention to research and study the porosity and permeability of natural rock salt. Rock salt in sedimentary basins has long been considered to be impermeable and provides a seal for hydrocarbon accumulations in geological structures. The low permeability of static rock salt is due to a percolation threshold. However, deformation may be able to overcome this threshold and allow fluid flow. We confirm the percolation threshold in static experiments on synthetic salt samples with X-ray microtomography. We then analyze wells penetrating salt deposits in the Gulf of Mexico. The observed hydrocarbon distributions in rock salt require that percolation occurred at porosities considerably below the static threshold, due to deformation-assisted percolation. In general, static percolation thresholds may not always limit fluid flow in deforming environments. Here we use pore-scale simulations of texturally equilibrated pore networks to study the possibility of core formation by porous flow in planetesimals. Rapid core formation in early planetary bodies is required by geochemical data from extinct radionuclides. The most obvious mechanism for metal-silicate differentiation is the segregation of dense core forming melts by porous flow. However, experimental observations show that the texturally equilibrated metallic melt resides in isolated pockets that prevent percolation towards the center. The proposed hypothesis in this dissertation is that the porosities can be large enough to exceed percolation threshold and allow metalic melt drainge to center. The melt network remains interconnected as drainage reduces the porosity below the percolation threshold and only 1-2% is trapped. X-ray microtomography of lodran-like meteorite NWA 2993 provides evidence that volume fraction of metallic phases can exceed this percolation threshold. Lattice Boltzmann simulations show that the permeability during drainage remains significant. A model for metal-silicate differentiation by porous flow in a viscously compacting planetesimal is also proposed and shows that the efficient core formation requires early accretion and is completed almost 2 Myr after the onset of melting.Show more Item Spatial stochastic models for network analysis(2019-08) Sankararaman, Abishek; Baccelli, F. (François), 1954-; deVeciana, Gustavo; Shakkottai, Sanjay; Dimakis, Alexandros; Neeman, JoesephShow more This thesis proposes new stochastic interacting particle models for networks, and studies some fundamental properties of these models. This thesis considers two application areas of networking - engineering design questions in future wireless systems and algorithmic tasks in large scale graph structured data. The key innovation introduced in this thesis is to bring tools and ideas from stochastic geometry to bear on the problems in both these application domains. We identify certain fundamental questions in design and engineering both wireless systems and large scale graph structured data processing systems. Subsequently, we identify novel stochastic geometric models, that captures the fundamental properties of these networks, which forms the first research contribution. We then rigorously study these models, by bringing to bear new tools from stochastic geometry, random graphs, percolation and Markov processes to establish structural results and fundamental phase transitions in these models. Using our developed mathematical methodology, we then identify design insights and develop algorithms, which we demonstrate are instructive in many practical settings. In the setting of wireless systems, this thesis studies both ad-hoc and cellular networks. In the ad-hoc network setting, we aim to understand fundamental limits of the simplest possible protocol to access the spectrum, namely a link transmits whenever it has data to send by treating all interference as noise. Surprisingly this basic question itself was not understood, as the system dynamics is coupled spatially due to the interference links cause one another and temporally due to randomness in traffic arrivals. We propose a novel interacting particle model called the spatial birth-death wireless network model to understand the stability properties of the simple spectrum access protocol. Using tools from Palm calculus and fluid limit theory, we establish a tight characterization of when this model is stable. Furthermore, we show that whenever stable, the links in steady-state exhibit a form of clustering. Leveraging these structural results, we propose two mean field heuristics to obtain formulas for key performance metrics such as average delay experienced by a link. We empirically find that the proposed formulas for delay predicts accurately the system behavior. We subsequently study scalability properties of this model by introducing an appropriate infinite dimensional version of the model we call the Interference Queueing Networks model. The model consists of a queue located at each grid point of an infinite regular integer lattice, with the queues interacting with each other in a translation invariant fashion. We then prove several structural properties of the model namely, tight conditions for existence of stationary solutions and some sufficient conditions for uniqueness of stationary solutions. Remarkably, we obtain exact formula for mean delay in this model, unlike the continuum model where we relied on mean-field type heuristics to obtain insights. In the setting of cellular networks, we study optimal association schemes by mobile phones in the case when there are several possible base station technologies operating on orthogonal bands. We show that this choice leads to a performance gain we term technology diversity. Interestingly, we show that the performance gain relies on the amount of instantaneous information a user has on the various base station technologies that it can leverage to make the association decision. We outline optimal association schemes under various information settings that a user may have on the network. Moreover, we propose simple heuristics for association that relies on a user obtaining minimal instantaneous information and are thus practical to implement. We prove that in certain natural asymptotic regime of parameters, our proposed heuristic policy is also optimal, and thus quantifying the value of having fine grained information at a user for association. We empirically observe that the asymptotic result is valid even at finite parameter regimes that are typical in todays networks. In the application of analyzing large scale graph structured data, we consider the graph clustering problem with side information. Graph clustering is a standard and widely used task which consists in partitioning the set of nodes of a graph into underlying clusters where nodes in the same cluster are similar to each other and nodes across different clusters are different. Motivated by applications in social and biological networks, we consider the task of clustering nodes of a graph, when there is side information on the nodes, other than that contained in the graph. For instance in social networks, one has access to meta data about a person (node in a social graph) such as age, location, income etc, along with the combinatorial data of who are his friends on the social graph. Similarly, in biological networks, there is often meta-data about an experiment that provides additional contextual data about a node, in addition to the combinatorial data. In this thesis, we propose a generative model for such graph structured data with side information, which is inspired by random graph models in stochastic geometry such as the random connection model and the generative models for networks with clusters without contexts, such as the stochastic block model or the planted partition model. We propose a novel graph model called the planted partition random connection model. Roughly speaking, in this model, each node has two labels - an observable R [superscript d] valued (for some fixed d) feature label and an unobservable binary valued community label. Conditional on the node labels, edges are drawn at random in this graph depending on both the feature and community labels of the two end points. The clustering task consists in recovering the underlying partition of nodes corresponding to the respective community labels better than a random assignment, when given an observation of the graph generated and the features of all nodes. We show that if the 'density of nodes', i.e., average number of nodes having features in an unit volume of space of R [superscript d] is small, then no algorithm can cluster the graph that can asymptotically beat a random assignment of community labels. On the contrary, if the density of nodes is sufficiently high, we give a simple algorithm that recovers the true underlying partition strictly better a random assignment. We then apply the proposed algorithm to a problem in computational biology called Haplotype Phasing and observe empirically, that it obtains state of art results. This demonstrates, both the validity of our generative model, as well as our new algorithm.Show more