Self-organization of network dynamics into local quantized states

Self-organization and pattern formation in network-organized systems emerges from the collective activation and interaction of many interconnected units. A striking feature of these non-equilibrium structures is that they are often localized and robust: only a small subset of the nodes, or cell assembly, is activated. Understanding the role of cell assemblies as basic functional units in neural networks and socio-technical systems emerges as a fundamental challenge in network theory. A key open question is how these elementary building blocks emerge, and how they operate, linking structure and function in complex networks. Here we show that a network analogue of the Swift-Hohenberg continuum model---a minimal-ingredients model of nodal activation and interaction within a complex network---is able to produce a complex suite of localized patterns. Hence, the spontaneous formation of robust operational cell assemblies in complex networks can be explained as the result of self-organization, even in the absence of synaptic reinforcements. Our results show that these self-organized, local structures can provide robust functional units to understand natural and socio-technical network-organized processes.Comment: 11 pages, 4 figure

Three-dimensional simulation of unstable gravity-driven infiltration of water into a porous medium

Infiltration of water in dry porous media is subject to a powerful gravity-driven instability. Although the phenomenon of unstable infiltration is well known, its description using continuum mathematical models has posed a significant challenge for several decades. The classical model of water flow in the unsaturated flow, the Richards equation, is unable to reproduce the instability. Here, we present a computational study of a model of unsaturated flow in porous media that extends the Richards equation and is capable of predicting the instability and captures the key features of gravity fingering quantitatively. The extended model is based on a phase-field formulation and is fourth-order in space. The new model poses a set of challenges for numerical discretizations, such as resolution of evolving interfaces, stiffness in space and time, treatment of singularly perturbed equations, and discretization of higher-order spatial partial–differential operators. We develop a numerical algorithm based on Isogeometric Analysis, a generalization of the finite element method that permits the use of globally-smooth basis functions, leading to a simple and efficient discretization of higher-order spatial operators in variational form. We illustrate the accuracy, efficiency and robustness of our method with several examples in two and three dimensions in both homogeneous and strongly heterogeneous media. We simulate, for the first time, unstable gravity-driven infiltration in three dimensions, and confirm that the new theory reproduces the fundamental features of water infiltration into a porous medium. Our results are consistent with classical experimental observations that demonstrate a transition from stable to unstable fronts depending on the infiltration flux.United States. Dept. of Energy (Early Career Award Grant DE-SC0003907

Viscous fingering with partially miscible fluids

Viscous fingering—the fluid-mechanical instability that takes place when a low-viscosity fluid displaces a high-viscosity fluid—has traditionally been studied under either fully miscible or fully immiscible fluid systems. Here we study the impact of partial miscibility (a common occurrence in practice) on the fingering dynamics. Through a careful design of the thermodynamic free energy of a binary mixture, we develop a phase-field model of fluid-fluid displacements in a Hele-Shaw cell for the general case in which the two fluids have limited (but nonzero) solubility into one another. We show, by means of high-resolution numerical simulations, that partial miscibility exerts a powerful control on the degree of fingering: fluid dissolution hinders fingering while fluid exsolution enhances fingering. We also show that, as a result of the interplay between compositional exchange and the hydrodynamic pattern-forming process, stronger fingering promotes the system to approach thermodynamic equilibrium more quickly

Pore-scale modeling of phase change in porous media

The combination of high-resolution visualization techniques and pore-scale flow modeling is a powerful tool used to understand multiphase flow mechanisms in porous media and their impact on reservoir-scale processes. One of the main open challenges in pore-scale modeling is the direct simulation of flows involving multicomponent mixtures with complex phase behavior. Reservoir fluid mixtures are often described through cubic equations of state, which makes diffuse-interface, or phase-field, theories particularly appealing as a modeling framework. What is still unclear is whether equation-of-state-driven diffuse-interface models can adequately describe processes where surface tension and wetting phenomena play important roles. Here we present a diffuse-interface model of single-component two-phase flow (a van der Waals fluid) in a porous medium under different wetting conditions. We propose a simplified Darcy-Korteweg model that is appropriate to describe flow in a Hele-Shaw cell or a micromodel, with a gap-averaged velocity. We study the ability of the diffuse-interface model to capture capillary pressure and the dynamics of vaporization-condensation fronts and show that the model reproduces pressure fluctuations that emerge from abrupt interface displacements (Haines jumps) and from the breakup of wetting films

Pattern formation and coarsening dynamics in three-dimensional convective mixing in porous media

Geological carbon dioxide (CO[subscript 2]) sequestration entails capturing and injecting CO[subscript 2][subscript 2]into deep saline aquifers for long-term storage. The injected CO[subscript 2] partially dissolves in groundwater to form a mixture that is denser than the initial groundwater. The local increase in density triggers a gravitational instability at the boundary layer that further develops into columnar plumes of CO[subscript 2]-rich brine, a process that greatly accelerates solubility trapping of the CO[subscript 2]. Here, we investigate the pattern-formation aspects of convective mixing during geological CO[subscript 2] sequestration by means of high-resolution three-dimensional simulation. We find that the CO[subscript 2] concentration field self-organizes as a cellular network structure in the diffusive boundary layer at the top boundary. By studying the statistics of the cellular network, we identify various regimes of finger coarsening over time, the existence of a non-equilibrium stationary state, and a universal scaling of three-dimensional convective mixing

A discrete-domain description of multiphase flow in porous media: rugged energy landscapes and the origin of hysteresis

We propose a discrete-domain model to describe mesoscale (many pore) immiscible displacements in porous media. We conceptualize the porous medium and fluid system as a set of weakly connected multistable compartments. The overall properties of the system emerge from the small-scale compartment dynamics. Our model aims at capturing the rugged energy landscape of multiphase porous media systems, emphasizing the role of metastability and local equilibria in the origin of hysteresis. Under two-phase displacements, the system behaves hysteretically, but our description does not rely on past saturations, turning points, or drainage/imbibition labels. We characterize the connection between micrometastability and overall system behavior, and elucidate the different nature of pressure-controlled and rate-controlled immiscible displacements in porous media

Phase field model of fluid-driven fracture in elastic media: Immersed-fracture formulation and validation with analytical solutions

Propagation of fluid-driven fractures plays an important role in natural and engineering processes, including transport of magma in the lithosphere, geologic sequestration of carbon dioxide, and oil and gas recovery from low-permeability formations, among many others. The simulation of fracture propagation poses a computational challenge as a result of the complex physics of fracture and the need to capture disparate length scales. Phase field models represent fractures as a diffuse interface and enjoy the advantage that fracture nucleation, propagation, branching, or twisting can be simulated without ad hoc computational strategies like remeshing or local enrichment of the solution space. Here we propose a new quasi-static phase field formulation for modeling fluid-driven fracturing in elastic media at small strains. The approach fully couples the fluid flow in the fracture (described via the Reynolds lubrication approximation) and the deformation of the surrounding medium. The flow is solved on a lower dimensionality mesh immersed in the elastic medium. This approach leads to accurate coupling of both physics. We assessed the performance of the model extensively by comparing results for the evolution of fracture length, aperture, and fracture fluid pressure against analytical solutions under different fracture propagation regimes. The excellent performance of the numerical model in all regimes builds confidence in the applicability of phase field approaches to simulate fluid-driven fracture.United States. Department of Energy (Grant DE-SC0009286)Spain. Ministerio de Economía y Competitividad (Grant RyC-2012-11704)Spain. Ministerio de Economía y Competitividad (Grant CTM2014-54312-P

Scaling of capillary trapping in unstable two-phase flow: Application to CO2 sequestration in deep saline aquifers

The effect of flow instabilities on capillary trapping mechanisms is a major source of uncertainty in CO2 sequestration in deep saline aquifers. Standard macroscopic models of multiphase flow in porous media are unable to explain and quantitatively predict the onset and structure of viscous-unstable flows, such as the displacement of brine by the injected CO2. We present the first step of a research effort aimed at the experimental characterization and mathematical (continuum) modeling of such flows. Existing continuum models of multiphase flow are unable to explain why preferential flow (fingering) occurs during infiltration into homogeneous, dry soil. We present a macroscopic model that reproduces the experimentally observed features of fingered flows. The proposed model is derived using a phase-field methodology and does not introduce new independent parameters. From a linear stability analysis, we predict that finger velocity and finger width both increase with infiltration rate, and the predictions are in quantitative agreement with experiments.Eni S.p.A. (Firm) (Multiscale Reservoir Science project)Massachusetts Institute of Technology. Department of Civil and Environmental Engineering (Gilbert Winslow Career Development Chair

Thermodynamic coarsening arrested by viscous fingering in partially miscible binary mixtures

We study the evolution of binary mixtures far from equilibrium, and show that the interplay between phase separation and hydrodynamic instability can arrest the Ostwald ripening process characteristic of nonflowing mixtures. We describe a model binary system in a Hele-Shaw cell using a phase-field approach with explicit dependence of both phase fraction and mass concentration. When the viscosity contrast between phases is large (as is the case for gas and liquid phases), an imposed background flow leads to viscous fingering, phase branching, and pinch off. This dynamic flow disorder limits phase growth and arrests thermodynamic coarsening. As a result, the system reaches a regime of statistical steady state in which the binary mixture is permanently driven away from equilibrium.United States. Dept. of Energy (CAREER Award DE-SC0003907)United States. Dept. of Energy (Grants DESC0009286 and DE-FE0013999)Spain. Ministerio de Economia y Competividad (Ramon y Cajal Fellowship

Sobre el empleo de técnicas de mínimos cuadrados móviles en el desarrollo de métodos de alta resolución de volúmenes finitos en mallas no estructuradas. Aplicación a problemas de flujo comprensible

Congreso de Métodos Numéricos en Ingeniería 2005, Granada, SpainEn este artículo se propone la combinación de aproximaciones por mínimos cuadrados móviles y esquemas upwind de volúmenes finitos, aplicados a problemas de flujo compresible en mallas no estructuradas. La construcción de métodos de alto orden de este tipo ha estado limitada por la necesidad de disponer de técnicas que permitan estimar las derivadas sucesivas de las variables de flujo a partir de sus valores promedio en las celdas. La utilización de MLS permite disponer de un marco de aproximación general, asi como desarrollar reconstrucciones de muy alto orden sin aumentar el número de grados de libertad del problema