The Parallel Complexity of Growth Models

This paper investigates the parallel complexity of several non-equilibrium growth models. Invasion percolation, Eden growth, ballistic deposition and solid-on-solid growth are all seemingly highly sequential processes that yield self-similar or self-affine random clusters. Nonetheless, we present fast parallel randomized algorithms for generating these clusters. The running times of the algorithms scale as $O(\log^2 N)$, where $N$ is the system size, and the number of processors required scale as a polynomial in $N$. The algorithms are based on fast parallel procedures for finding minimum weight paths; they illuminate the close connection between growth models and self-avoiding paths in random environments. In addition to their potential practical value, our algorithms serve to classify these growth models as less complex than other growth models, such as diffusion-limited aggregation, for which fast parallel algorithms probably do not exist.Comment: 20 pages, latex, submitted to J. Stat. Phys., UNH-TR94-0

The shape of invasion perclation clusters in random and correlated media

The shape of two-dimensional invasion percolation clusters are studied numerically for both non-trapping (NTIP) and trapping (TIP) invasion percolation processes. Two different anisotropy quantifiers, the anisotropy parameter and the asphericity are used for probing the degree of anisotropy of clusters. We observe that in spite of the difference in scaling properties of NTIP and TIP, there is no difference in the values of anisotropy quantifiers of these processes. Furthermore, we find that in completely random media, the invasion percolation clusters are on average slightly less isotropic than standard percolation clusters. Introducing isotropic long-range correlations into the media reduces the isotropy of the invasion percolation clusters. The effect is more pronounced for the case of persisting long-range correlations. The implication of boundary conditions on the shape of clusters is another subject of interest. Compared to the case of free boundary conditions, IP clusters of conventional rectangular geometry turn out to be more isotropic. Moreover, we see that in conventional rectangular geometry the NTIP clusters are more isotropic than TIP clusters

Percolation with trapping mechanism drives active gels to the critically connected state

Cell motility and tissue morphogenesis depend crucially on the dynamic remodelling of actomyosin networks. An actomyosin network consists of an actin polymer network connected by crosslinker proteins and motor protein myosins that generate internal stresses on the network. A recent discovery shows that for a range of experimental parameters, actomyosin networks contract to clusters with a power-law size distribution [Alvarado J. et al. (2013) Nature Physics 9 591]. Here, we argue that actomyosin networks can exhibit robust critical signature without fine-tuning because the dynamics of the system can be mapped onto a modified version of percolation with trapping (PT), which is known to show critical behaviour belonging to the static percolation universality class without the need of fine-tuning of a control parameter. We further employ our PT model to generate experimentally testable predictions.Comment: 7 pages, 6 figures. To appear in Physical Review

Capillary Hysteresis in Neutrally Wettable Fibrous Media: A Pore Network Study of a Fuel Cell Electrode

Hysteresis in the saturation versus capillary pressure curves of neutrally wettable fibrous media was simulated with a random pore network model using a Voronoi diagram approach. The network was calibrated to fit experimental air-water capillary pressure data collected for carbon fibre paper commonly used as a gas diffusion layer in fuel cells. These materials exhibit unusually strong capillary hysteresis, to the extent that water injection and withdrawal occur at positive and negative capillary pressures, respectively. Without the need to invoke contact angle hysteresis, this capillary behaviour is re-produced when using a pore-scale model based on the curvature of a meniscus passing through the centre of a toroid. The classic Washburn relation was shown to produce erroneous results, and its use is not recommended when modelling fibrous media. The important effect of saturation distribution on the effective diffusivity of the medium was also investigated for both water injection and withdrawal cases. The findings have bearing on the understanding of both capillarity in fibrous media and fuel cell design

Two-phase flow in rocks : new insights from multi-scale pore network modeling and fast pore scale visualization

Many geological applications involve the flow of multiple fluids through porous geological materials, e.g. environmental remediation of polluted ground water resources, carbon dioxide storage in geological reservoirs and petroleum recovery. Commonly, to model these applications, the geological materials in question are treated as continuous porous media with effective material properties. Since these properties are a manifestation of what goes on in the pores of the material, we have to study the transport processes at the pore scale to understand why and how they vary over space and time in different rocks and under different conditions. As the high cost of acquiring and testing samples in many of these applications is often a limiting factor, numerical modelling at the pore scale is becoming a key technology to gain new insights in this field. This could be crucial in reducing uncertainties in field scale projects. The work presented in this thesis focuses on the investigation of two-phase flow in sedimentary rocks, and is an integrated numerical and experimental study. It deals primarily with two outstanding issues. First, image-based pore scale simulation methods have difficulties with representing the multiple pore scales in rocks with wide pore size distributions, due to a trade-off in the size and resolution of both modeling and imaging methods. Therefore, performing two-phase flow simulations in a number of important rock types, such as many carbonates and tight, clay-baring sandstones has remained an outstanding challenge. To tackle this problem, a new numerical model was developed to calculate capillary pressure, relative permeability and resistivity index curves during drainage and imbibition processes in such materials. The multi-scale model was based on information obtained from 3D micro-computed tomography images of the internal pore structure, complemented with information on the pores that are unresolved with this technique. In this method, pore network models were first extracted from resolved pores in the images, by using a maximal ball network extraction algorithm. Then, pores which touched regions with unresolved porosity were connected with a special type of network element called micro-links. In the quasi-static simulations that were performed on these network models, the micro-links carried average properties of the unresolved porosity. In contrast to most previous models, the new approach to taking into account unresolved porosity therefore allowed efficient simulations on images of complex rocks, with sizes comparable to single-scale pore network models. It was able to reproduce most of the behaviour of a fully resolved pore network model, for both drainage and imbibition processes, and for different pore scale wettability distributions (water-wet, oil-wet and different mixed-wet distributions). Furthermore, simulations on images of carbonate rocks showed good agreement to experiments. A sensitivity study on carbonate rocks and tight, clay-bearing sandstones produced results that were in qualitative agreement with experiments, and allowed to analyse how the two-phase flow behaviour of these rocks is influenced by their pore scale properties. The second issue which is treated in this thesis is related to the validation of pore scale models. Comparing predicted effective properties to experimentally measured values is useful and necessary, but is complicated by the typical difference in size between the model and the experiment. Furthermore, it does not always give a clear indication of the reasons for an observed mismatch between models and experiments. Comparing two-phase flow models to pore scale experiments in which the evolution of the fluid distributions is visualized is thus extremely useful. However, this requires to image the two-phase flow process while it is taking place in a rock, and it is necessary to do this with time resolutions on the order of tens of seconds and spatial resolutions on the order of micrometers. Previous experimental approaches used synchrotron beam lines to achieve this. In this thesis, we show that such experiments are also possible using laboratory-based micro-computed tomography scanners, which are orders of magnitude cheaper and therefore more accessible than synchrotrons. An experiment in which kerosene was pumped into a water-saturated sandstone is presented, showing that individual Haines jumps (pore filling events) could be visualized during this drainage process. Because the image quality is lower than at synchrotrons, care had to be taken to adapt the image analysis work flow to deal with high image noise levels. The work flow was designed to allow to track the fluid filling state of individual pores. The results indicate that the dynamic effects due to viscous and inertial forces during Haines jumps do not significantly impact the evolution of the fluid distributions during drainage, which may thus be adequately described by quasi-static models

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 non-wetting phases that evolve before steady-state has occurred, undergo a cross-over where every initial patterns are broken up. For flow dominated by capillary effects with capillary numbers in order of $10^{-5}$, we find that around a critical saturation of non-wetting fluid the non-wetting clusters of size $s$ have a power-law distribution $n_s \sim s^{-\tau}$ with the exponent $\tau = 1.92 \pm 0.04$ 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.Comment: 12 pages, 11 figures. Minor corrections. Accepted for publication in Phys. Rev.

Local capillary trapping and permeability-retarded accumulation during geologic carbon sequestration

Safe storage of CO2 in saline aquifers depends on CO2 migration rate, accumulation, and trapping inside saline aquifers that have intrinsic heterogeneity. This heterogeneity can be in both capillary entry pressure and permeability. The former heterogeneity causes local capillary trapping while the latter results in permeability-retarded accumulation. A main objective of this dissertation is to understand how both local capillary trapping and permeability-retarded accumulation secure CO2 storage. We establish a fast simulation technique to model local capillary trapping during CO2 injection into saline aquifers. In this technique, modeling efforts are decoupled into two parts: identifying trapping in a capillary entry pressure field and simulating CO2 flow in a permeability field. The former fields are correlated with the latter using the Leverett j-function. The first part describes an extended use of a geologic criterion originally proposed by Saadatpoor (2012). This criterion refers to a single value of ‘critical capillary entry pressure’ that is used to indicate barrier or local traps cells during buoyant flow. Three issues with the criterion are the unknown physical critical value, the massive overestimation of trapping, and boundary barriers. The first two issues are resolved through incorporating viscous flow of CO2. The last issue is resolved through creating periodic boundaries. This creation enables us to study both the amount and clusters of local capillary traps in infinite systems, and meanwhile the effects of reservoir heterogeneity, system size, aspect ratio, and boundary types are examined. In the second part, we adapt a connectivity analysis to assess CO2 plume dynamics. This analysis is then integrated into the geologic criterion to evaluate how injection strategies affect local capillary trapping in reservoirs. We demonstrate that reservoir heterogeneity affects the optimal injection strategies in terms of maximizing this trapping. We conduct analytical and numerical modeling of CO2 accumulations caused by both permeability hindrances and capillary barriers. The analytical model describes CO2 buoyant migration and accumulation at a low permeability region above a high-permeability region. In the limiting case of zero capillary pressure, the model equation is solved using the method of characteristics. The permeability-retarded accumulation is illustrated through CO2 saturation profiles and time-distance diagrams. Capillary trapping is subsequently accounted for by graphically incorporating the capillary pressure curve and capillary threshold effect. The relative importance of these two types of accumulations is examined under various buoyant source fluxes and porous media properties. Results demonstrate that accumulation estimate that account for only capillary trapping understates the amount of CO2 accumulated beneath low permeability structures during significant periods of a sequestration operation.Petroleum and Geosystems Engineerin

A gravity current model with capillary trapping for oil migration in multilayer geological basins

We propose a reduced model accounting capillary trapping to simulate oil migration in geological basins made of several rock types. Our model is derived from Darcy type models thanks to Dupuit approximation and a vertical integration in each geological layer. We propose a time-implicit finite volume scheme which is shown to be unconditionally stable and to admit discrete solutions. Numerical outcomes are then provided in order to illustrate the behavior of our reduced model