Publications

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Competition is fierce and often the first to act has an advantage, especially in environments where there are excess resources. However, expanding quickly to absorb excess resources creates requirements that might be unmet in future conditions of scarcity. Different patterns of scarcity call for different strategies. We define a model of interacting specialists (entities) to analyze which sizing strategies are most successful in environments subjected to frequent periods of scarcity. We require entities to compete for a common resource whose scarcity changes periodically, then study the viability of entities following three different strategies through scarcity episodes of varying duration and intensity. The three sizing strategies are: aggressive, moderate, and conservative. Aggressive strategies are most effective when the episodes of scarcity are shorter and moderate; conversely, conservative strategies are most effective in cases of longer or more severe scarcity. © 2012 Springer-Verlag.

Complex adaptive systems (CAS) modeling has become a common tool to study the behavioral dynamics of agents in a broad range of disciplines from ecology to economics. Many modelers have studied structure's importance for a system in equilibrium, while others study the effects of perturbations on system dynamics. There is a notable absence of work on the effects of agent interaction pathways on perturbation dynamics. We present an agent-based CAS model of a competitive economic environment. We use this model to study the perturbation dynamics of simple structures by introducing a series of disruptive events and observing key system metrics. Then, we generate more complex networks by combining the simple component structures and analyze the resulting dynamics. We find the local network structure of a perturbed node to be a valuable indicator of the system response. © 2012 Springer-Verlag.

International forces in Afghanistan have experienced difficulties in developing constructive engagements with the Afghan population, an experience familiar to a wide range of international agencies working in underdeveloped and developing nations around the world. Recently, forces have begun deploying Female Engagement Teams, female military units who engage directly with women in occupied communities, resulting inmore positive relationships with those communities as a whole. In this paper, we explore the hypothesis that the structure of community-based social networks strongly contributes to the effectiveness of the Female Engagement Team strategy, specifically considering gender-based differences in network community structure. We find that the ability to address both female and male network components provides a superior ability to affect opinions in the network, and can provide an effective means of counteracting influences from opposition forces. © 2012 Springer-Verlag.

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Complex Adaptive Systems of Systems, or CASoS, are vastly complex physical-socio-technical systems which we must understand to design a secure future for the nation. The Phoenix initiative implements CASoS Engineering principles combining the bottom up Complex Systems and Complex Adaptive Systems view with the top down Systems Engineering and System-of-Systems view. CASoS Engineering theory and practice must be conducted together to develop a discipline that is grounded in reality, extends our understanding of how CASoS behave and allows us to better control the outcomes. The pull of applications (real world problems) is critical to this effort, as is the articulation of a CASoS Engineering Framework that grounds an engineering approach in the theory of complex adaptive systems of systems. Successful application of the CASoS Engineering Framework requires modeling, simulation and analysis (MS and A) capabilities and the cultivation of a CASoS Engineering Community of Practice through knowledge sharing and facilitation. The CASoS Engineering Environment, itself a complex adaptive system of systems, constitutes the two platforms that provide these capabilities.

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Infrastructures are networks of dynamically interacting systems designed for the flow of information, energy, and materials. Under certain circumstances, disturbances from a targeted attack or natural disasters can cause cascading failures within and between infrastructures that result in significant service losses and long recovery times. Reliable interdependency models that can capture such multi-network cascading do not exist. The research reported here has extended Sandia's infrastructure modeling capabilities by: (1) addressing interdependencies among networks, (2) incorporating adaptive behavioral models into the network models, and (3) providing mechanisms for evaluating vulnerability to targeted attack and unforeseen disruptions. We have applied these capabilities to evaluate the robustness of various systems, and to identify factors that control the scale and duration of disruption. This capability lays the foundation for developing advanced system security solutions that encompass both external shocks and internal dynamics.

Complex Adaptive Systems of Systems, or CASoS, are vastly complex eco-socio-economic-technical systems which we must understand to design a secure future for the nation and the world. Perturbations/disruptions in CASoS have the potential for far-reaching effects due to highly-saturated interdependencies and allied vulnerabilities to cascades in associated systems. The Phoenix initiative approaches this high-impact problem space as engineers, devising interventions (problem solutions) that influence CASoS to achieve specific aspirations. CASoS embody the world's biggest problems and greatest opportunities: applications to real world problems are the driving force of our effort. We are developing engineering theory and practice together to create a discipline that is grounded in reality, extends our understanding of how CASoS behave, and allows us to better control those behaviors. Through application to real-world problems, Phoenix is evolving CASoS Engineering principles while growing a community of practice and the CASoS engineers to populate it.

Complex Adaptive Systems of Systems, or CASoS, are vastly complex ecological, sociological, economic and/or technical systems which we must understand to design a secure future for the nation and the world. Perturbations/disruptions in CASoS have the potential for far-reaching effects due to pervasive interdependencies and attendant vulnerabilities to cascades in associated systems. Phoenix was initiated to address this high-impact problem space as engineers. Our overarching goals are maximizing security, maximizing health, and minimizing risk. We design interventions, or problem solutions, that influence CASoS to achieve specific aspirations. Through application to real-world problems, Phoenix is evolving the principles and discipline of CASoS Engineering while growing a community of practice and the CASoS engineers to populate it. Both grounded in reality and working to extend our understanding and control of that reality, Phoenix is at the same time a solution within a CASoS and a CASoS itself.

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The objectives of this presentation are: (1) To determine if healthcare settings serve as intensive transmission environments for influenza epidemics, increasing effects on communities; (2) To determine which mitigation strategies are best for use in healthcare settings and in communities to limit influenza epidemic effects; and (3) To determine which mitigation strategies are best to prevent illness in healthcare workers.

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Cigarette smoking presented the most significant public health challenge in the United States in the 20th Century and remains the single most preventable cause of morbidity and mortality in this country. A number of System Dynamics models exist that inform tobacco control policies. We reviewed them and discuss their contributions. We developed a theory of the societal lifecycle of smoking, using a parsimonious set of feedback loops to capture historical trends and explore future scenarios. Previous work did not explain the long-term historical patterns of smoking behaviors. Much of it used stock-and-flow to represent the decline in prevalence in the recent past. With noted exceptions, information feedbacks were not embedded in these models. We present and discuss our feedback-rich conceptual model and illustrate the results of a series of simulations. A formal analysis shows phenomena composed of different phases of behavior with specific dominant feedbacks associated with each phase. We discuss the implications of our society's current phase, and conclude with simulations of what-if scenarios. Because System Dynamics models must contain information feedback to be able to anticipate tipping points and to help identify policies that exploit leverage in a complex system, we expanded this body of work to provide an endogenous representation of the century-long societal lifecycle of smoking.

Loki-Infect 3 is a desktop application intended for use by community-level decision makers. It allows rapid construction of small-scale studies of emerging or hypothetical infectious diseases in their communities and evaluation of the potential effectiveness of various containment strategies. It was designed with an emphasis on modularity, portability, and ease of use. Our goal is to make this program freely available to community workers across the world.

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We develop a parsimonious model of the interbank payment system. The model incorporates an endogenous instruction arrival process, a scale-free topology of payments between banks, a fixed total liquidity which limits banks' capacity to process arriving instructions, and a global market that distributes liquidity. We find that at low liquidity the system becomes congested and payment settlement loses correlation with payment instruction arrival, becoming coupled across the network. The onset of congestion is evidently related to the relative values of three characteristic times: the time for banks' net position to return to 0, the time for a bank to exhaust its liquidity endowment, and the liquidity market relaxation time. In the congested regime settlement takes place in cascades having a characteristic length scale. A global liquidity market substantially attenuates congestion, requiring only a small fraction of the payment-induced liquidity flow to achieve strong beneficial effects. © 2007 Elsevier B.V. All rights reserved.

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This report describes a model Transport Processes Investigation (TPI) where field-scale vadose zone flow and transport processes are identified and verified through a systematic field investigation at a contaminated DOE site. The objective of the TPI is to help with formulating accurate conceptual models and aid in implementing rational and cost effective site specific characterization strategies at contaminated sites with diverse hydrogeologic settings. Central to the TPI are Transport Processes Characterization (TPC) tests that incorporate field surveys and large-scale infiltration experiments. Hypotheses are formulated based on observed pedogenic and hydrogeologic features as well as information provided by literature searches. The field and literature information is then used to optimize the design of one or more infiltration experiments to field test the hypothesis. Findings from the field surveys and infiltration experiments are then synthesized to formulate accurate flow and transport conceptual models. Here we document a TPI implemented in the glacial till vadose zone at the Fernald Environmental Management Project (FEMP) in Fernald, Ohio, a US Department of Energy (DOE) uranium processing site. As a result of this TPI, the flow and transport mechanisms were identified through visualization of dye stain within extensive macro pore and fracture networks which provided the means for the infiltrate to bypass potential aquatards. Such mechanisms are not addressed in current vadose zone modeling and are generally missed by classical characterization methods.

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An infiltration and dye transport experiment was conducted to visualize flow and transport processes in a heterogeneous, layered, sandy-gravelly fluvial deposit adjacent to Rio Bravo Boulevard in Albuquerque, NM. Water containing red dye followed by blue-green dye was ponded in a small horizontal zone ({approx}0.5 m x 0.5 m) above a vertical outcrop ({approx}4 m x 2.5 m). The red dye lagged behind the wetting front due to slight adsorption thus allowing both the wetting front and dye fronts to be observed in time at the outcrop face. After infiltration, vertical slices were excavated to the midpoint of the infiltrometer exposing the wetting front and dye distribution in a quasi three-dimensional manner. At small-scale, wetting front advancement was influenced by the multitude of local capillary barriers within the deposit. However at the scale of the experiment, the wetting front appeared smooth with significant lateral spreading {approx} twice that in the vertical, indicating a strong anisotropy due to the pronounced horizontal layering. The dye fronts exhibited appreciably more irregularity than the wetting front, as well as the influence of preferential flow features (a fracture) that moved the dye directly to the front, bypassing the fresh water between.

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Concepts from Complexity Science are valuable and allow a simulation approach for critical infrastructures that is flexible and has wide ranging applications.

Critical Infrastructures are formed by a large number of components that interact within complex networks. As a rule, infrastructures contain strong feedbacks either explicitly through the action of hardware/software control, or implicitly through the action/reaction of people. Individual infrastructures influence others and grow, adapt, and thus evolve in response to their multifaceted physical, economic, cultural, and political environments. Simply put, critical infrastructures are complex adaptive systems. In the Advanced Modeling and Techniques Investigations (AMTI) subgroup of the National Infrastructure Simulation and Analysis Center (NISAC), we are studying infrastructures as complex adaptive systems. In one of AMTI's efforts, we are focusing on cascading failure as can occur with devastating results within and between infrastructures. Over the past year we have synthesized and extended the large variety of abstract cascade models developed in the field of complexity science and have started to apply them to specific infrastructures that might experience cascading failure. In this report we introduce our comprehensive model, Polynet, which simulates cascading failure over a wide range of network topologies, interaction rules, and adaptive responses as well as multiple interacting and growing networks. We first demonstrate Polynet for the classical Bac, Tang, and Wiesenfeld or BTW sand-pile in several network topologies. We then apply Polynet to two very different critical infrastructures: the high voltage electric power transmission system which relays electricity from generators to groups of distribution-level consumers, and Fedwire which is a Federal Reserve service for sending large-value payments between banks and other large financial institutions. For these two applications, we tailor interaction rules to represent appropriate unit behavior and consider the influence of random transactions within two stylized networks: a regular homogeneous array and a heterogeneous scale-free (fractal) network. For the stylized electric power grid, our initial simulations demonstrate that the addition of geographically unrestricted random transactions can eventually push a grid to cascading failure, thus supporting the hypothesis that actions of unrestrained power markets (without proper security coordination on market actions) can undermine large scale system stability. We also find that network topology greatly influences system robustness. Homogeneous networks that are 'fish-net' like can withstand many more transaction perturbations before cascading than can scale-free networks. Interestingly, when the homogeneous network finally cascades, it tends to fail in its entirety, while the scale-free tends to compartmentalize failure and thus leads to smaller, more restricted outages. In the case of stylized Fedwire, initial simulations show that as banks adaptively set their individual reserves in response to random transactions, the ratio of the total volume of transactions to individual reserves, or 'turnover ratio', increases with increasing volume. The removal of a bank from interaction within the network then creates a cascade, its speed of propagation increasing as the turnover ratio increases. We also find that propagation is accelerated by patterned transactions (as expected to occur within real markets) and in scale-free networks, by the 'attack' of the most highly connected bank. These results suggest that the time scale for intervention by the Federal Reserve to divert a cascade in Fedwire may be quite short. Ongoing work in our cascade analysis effort is building on both these specific stylized applications to enhance their fidelity as well as embracing new applications. We are implementing markets and additional network interactions (e.g., social, telecommunication, information gathering, and control) that can impose structured drives (perturbations) comparable to those seen in real systems. Understanding the interaction of multiple networks, their interdependencies, and in particular, the underlying mechanisms for their growth/evolution is paramount. With this understanding, appropriate public policy can be identified to guide the evolution of present infrastructures to withstand the demands and threats of the future.

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We consider the influence of ambient groundwater flow on the migration of DNAPL within a fracture network. In context of a modified invasion percolation (MIP) growth algorithm, we formulate a mechanistic model that includes capillary and gravity forces as well as viscous forces within the DNAPL and the ambient groundwater. The MIP model is verified against laboratory experiments, which show good agreement in DNAPL migration path through a two-dimensional fracture network. The results of both simulations and laboratory experiments suggest that ambient groundwater flow can be a significant factor controlling DNAPL migration path, velocity, and channeling pattern in a fracture network.

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Fingering, nonmonotonicity, fragmentation, and pulsation within gravity/buoyant destabilized two-phase/unsaturated flow systems has been widely observed with examples in homogeneous to heterogeneous porous media, in single fractures to fracture networks, and for both wetting and nonwetting invasion. To model this phenomena, we consider a mechanistic approach based on forms of modified invasion percolation (MIP) that include gravity, the influence of the local interfacial curvature along the phase-phase interface, and the simultaneous invasion and reinvasion of both wetting and nonwetting fluids. We present example simulations and compare them to experimental data for three very different situations: (1) downward gravity-driven fingering of water into a dry, homogeneous, water-wettable, porous medium; (2) upward buoyancy-driven migration of gas within a water saturated, heterogeneous, water-wettable, porous medium; and (3) downward gravity-driven fingering of water into a dry, water-wettable, rough-walled fracture.

We consider the use of a hypodiffusive governing equation (HDE) for the porous-continuum modeling of gravity-driven fingers (GDF) as occur in initially dry, highly nonlinear, and hysteretic porous media. In addition to the capillary and gravity terms within the traditional Richards equation, the HDE contains a hypodiffusive term that models an experimentally observed hold-back-pile-up (HBPU) effect and thus imparts nonmonotonicity at the wetting front. In its dimensionless form the HDE contains the dimensionless hypodiffusion number, NHD. As NHD increases, one-dimensional (1D) numerical solutions transition from monotonic to nonmonotonic. Considering the experimentally observed controls on GDF occurrence, as either the initial moisture content and applied flux increase or the material nonlinearity decreases, solutions undergo the required transition back to monotonic. Additional tests for horizontal imbibition and capillary rise show the HDE to yield the required monotonie response but display sharper fronts for NHD > 0. Finally, two-dimensional (2D) numerical solutions illustrate that in parameter space where the 1D HDE yields nonmonotonicity, in 2D it forms nonmonotonic GDF.

A simple experiment finds that fracture intersections can act to integrate unsaturated flows, such that regular, low flows entering the intersection from above are transformed into large, less frequent pulses below. At low flows, our simple intersection forms two capillary barriers. Water from above pools at the intersection until sufficient pressure builds to breach the barriers and discharge stored fluid. The barriers then reform and the process is repeated. At low flows, the volume discharged from the intersection remains relatively uniform across a range of flow rates. At higher flows, discharge volume is highly variable, and a viscous stabilized non-pulsating regime occurs at the highest flow rate that we considered.

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Methods to determine unsaturated hydraulic properties can exhibit random and nonunique behavior. We assess the causes for these behaviors by visualizing microscale phase displacement processes that occur during equilibrium retention and transient outflow experiments. For both types of experiments we observe the drainage process to be composed of a mixture of fast air fingering and slower air back-filling. The influence of each of these microscale processes is controlled by a combination of the size and the speed of the applied boundary step, the initial saturation and its structure, and small-scale heterogeneities. Because the mixture of these microscale processes yields macroscale effective behavior, measured unsaturated flow properties are also a function of these controls. Such results suggest limitations on the current definitions and uniqueness of unsaturated hydraulic properties.

Hydraulic property measurements often rely on non-linear inversion models whose errors vary between samples. In non-linear physical measurement systems, bias can be directly quantified and removed using calibration standards. In hydrologic systems, field calibration is often infeasible and bias must be quantified indirectly. We use a Monte Carlo error analysis to indirectly quantify spatial bias in the saturated hydraulic conductivity, K{sub s}, and the exponential relative permeability parameter, {alpha}, estimated using a tension infiltrometer. Two types of observation error are considered, along with one inversion-model error resulting from poor contact between the instrument and the medium. Estimates of spatial statistics, including the mean, variance, and variogram-model parameters, show significant bias across a parameter space representative of poorly- to well-sorted silty sand to very coarse sand. When only observation errors are present, spatial statistics for both parameters are best estimated in materials with high hydraulic conductivity, like very coarse sand. When simple contact errors are included, the nature of the bias changes dramatically. Spatial statistics are poorly estimated, even in highly conductive materials. Conditions that permit accurate estimation of the statistics for one of the parameters prevent accurate estimation for the other; accurate regions for the two parameters do not overlap in parameter space. False cross-correlation between estimated parameters is created because estimates of K{sub s} also depend on estimates of {alpha} and both parameters are estimated from the same data.

Double-diffusive finger convection is a hydrodynamic instability that can occur when two components with different diffusivities are oppositely stratified with respect to the fluid density gradient as a critical condition is exceeded. Laboratory experiments were designed using sodium chloride and sucrose solutions in a Hele-Shaw cell. A high resolution, full field, light transmission technique was used to study the development of the instability. The initial buoyancy ratio (R{sub p}), which is a ratio of fluid density contributions by the two solutes, was varied systematically in the experiments so that the range of parameter space spanned conditions that were nearly stable (R{sub p} = 2.8) to those that were moderately unstable (R{sub p} = 1.4). In systems of low R{sub p}, fingers develop within several minutes, merge with adjacent fingers, form conduits, and stall before newer-generated fingers travel through the conduits and continue the process. Solute fluxes in low R{sub p} systems quickly reach steady state and are on the order of 10{sup {minus}6} m{sup 2} sec{sup {minus}1}. In the higher R{sub p} experiments, fingers are slower to evolve and do not interact as dynamically as in the lower R{sub p} systems. Our experiment with initial R{sub p} = 2.8 exhibited flux on the order of that expected for a similar diffusive system (i.e., 10{sup {minus}7} m{sup 2} sec{sup {minus}1}), although the structures were very different than the pattern of transport expected in a diffusing system. Mass flux decayed as t{sup 1/2} in two experiments each with initial R{sub p} = 2.4 and 2.8.

An experimental investigation was conducted to study double-diffusive finger convection in a Hele-Shaw cell by layering a sucrose solution over a more-dense sodium chloride (NaCl) solution. The solutal Rayleigh numbers were on the order of 60,000, based upon the height of the cell (25 cm), and the buoyancy ratio was 1.2. A full-field light transmission technique was used to measure a dye tracer dissolved in the NaCl solution. They analyze the concentration fields to yield the temporal evolution of length scales associated with the vertical and horizontal finger structure as well as the mass flux. These measures show a rapid progression through two early stages to a mature stage and finally a rundown period where mass flux decays rapidly. The data are useful for the development and evaluation of numerical simulators designed to model diffusion and convection of multiple components in porous media. The results are useful for correct formulation at both the process scale (the scale of the experiment) and effective scale (where the lab-scale processes are averaged-up to produce averaged parameters). A fundamental understanding of the fine-scale dynamics of double-diffusive finger convection is necessary in order to successfully parameterize large-scale systems.

The authors performed a simple experiment to elucidate phase structure within a pervasively fractured welded tuff. Dyed water was infiltrated from a surface pond over a 36 minute period while a geophysical array monitored the wetted region within vertical planes directly beneath. They then excavated the rock mass to a depth of {approximately}5 m and mapped the fracture network and extent of dye staining in a series of horizontal pavements. Near the pond the network was fully stained. Below, the phase structure immediately expanded and with depth, the structure became fragmented and complicated exhibiting evidence of preferential flow, fingers, irregular wetting patterns, and varied behavior at fracture intersections. Limited transient geophysical data suggested that strong vertical pathways form first followed by increased horizontal expansion and connection within the network. These rapid pathways are also the first to drain. Estimates also suggest that the excavation captured from {approximately}10% to 1% or less of the volume of rock interrogated by the infiltration slug and thus the penetration depth could have been quite large.

The authors reconceptualize macro modified invasion percolation (MMIP) at the near pore (NP) scale and apply it to simulate the non-wetting phase invasion experiments of Glass et al [in review] conducted in macro-heterogeneous porous media. For experiments where viscous forces were non-negligible, they redefine the total pore filling pressure to include viscous losses within the invading phase as well as the viscous influence to decrease randomness imposed by capillary forces at the front. NP-MMIP exhibits the complex invasion order seen experimentally with characteristic alternations between periods of gravity stabilized and destabilized invasion growth controlled by capillary barriers. The breaching of these barriers and subsequent pore scale fingering of the non-wetting phase is represented extremely well as is the saturation field evolution, and total volume invaded.

Surfactant-enhanced aquifer remediation is an emerging technology for aquifers contaminated with nonaqueous phase liquids (NAPLs). A two-dimensional micromodel and image capture system were applied to observe NAPL mobilization and solubilization phenomena. In each experiment, a common residual NAPL field was established, followed by a series of mobilization and solubilization experiments. Mobilization floods included pure water floods with variable flow rates and surfactant floods with variations in surfactant formulations. At relatively low capillary numbers (N{sub ca}<10{sup {minus}3}), the surfactant mobilization floods resulted in higher NAPL saturations than for the pure water flood, for similar N{sub ca}.These differences in macroscopic saturations are explained by differences in micro-scale mobilization processes. Solubilization of the residual NAPL remaining after the mobilization stage was dominated by the formation of dissolution fingers, which produced nonequilibrium NAPL solubilization. A macroemulsion phase also as observed to form spontaneously and persist during the solubilization stage of the experiments.

The authors consider the ability of the numerical solution of Richards equation to model gravity-driven fingers. Although gravity-driven fingers can be easily simulated using a partial downwind averaging method, they find the fingers are purely artificial, generated by the combined effects of truncation error induced oscillations and capillary hysteresis. Since Richards equation can only yield a monotonic solution for standard constitutive relations and constant flux boundary conditions, it is not the valid governing equation to model gravity-driven fingers, and therefore is also suspect for unsaturated flow in initially dry, highly nonlinear, and hysteretic media where these fingers occur. However, analysis of truncation error at the wetting front for the partial downwind method suggests the required mathematical behavior of a more comprehensive and physically based modeling approach for this region of parameter space.

Recent work demonstrates that phase displacements within horizontal fractures large with respect to the spatial correlation length of the aperture field lead to a satiated condition that constrains the relative permeability to be less than one. The authors use effective media theory to develop a conceptual model for satiated relative permeability, then compare predictions to existing experimental measurements, and numerical solutions of the Reynolds equation on the measured aperture field within the flowing phase. The close agreement among all results and data show that for the experiments considered here, in-plane tortuosity induced by the entrapped phase is the dominant factor controlling satiated relative permeability. They also find that for this data set, each factor in the conceptual model displays an approximate power law dependence on the satiated saturation of the fracture.

Fracture transmissivity and detailed aperture fields are measured in analog fractures specially designed to evaluate the utility of the Reynolds equation. The authors employ a light transmission technique with well-defined accuracy ({approximately}1% error) to measure aperture fields at high spatial resolution ({approximately}0.015 cm). A Hele-Shaw cell is used to confirm the approach by demonstrating agreement between experimental transmissivity, simulated transmissivity on the measured aperture field, and the parallel plate law. In the two rough-walled analog fractures considered, the discrepancy between the experimental and numerical estimates of fracture transmissivity was sufficiently large ({approximately} 22--47%) to exclude numerical and experimental errors (< 2%)as a source. They conclude that the three-dimensional character of the flow field is important for fully describing fluid flow in the two rough-walled fractures considered, and that the approach of depth averaging inherent in the formulation of the Reynolds equation is inadequate. They also explore the effects of spatial resolution, aperture measurement technique, and alternative definitions for link transmissivities in the finite-difference formulation, including some that contain corrections for tortuosity perpendicular to the mean fracture plane and Stokes flow. Various formulations for link transmissivity are shown to converge at high resolution ({approximately} 1/5 the spatial correlation length) in the smoothly varying fracture. At coarser resolutions, the solution becomes increasingly sensitive to definition of link transmissivity and measurement technique. Aperture measurements that integrate over individual grid blocks were less sensitive to measurement scale and definition of link transmissivity than point sampling techniques.

The authors develop and evaluate a modified invasion percolation (MIP) model for quasi-static immiscible displacement in horizontal fractures. The effects of contact angle, local aperture field geometry, and local in-plane interracial curvature between phases are included in the calculation of invasion pressure for individual sites in a discretized aperture field. This pressure controls the choice of which site is invaded during the displacement process and hence the growth of phase saturation structure within the fracture. To focus on the influence of local in-plane curvature on phase invasion structure, they formulate a simplified nondimensional pressure equation containing a dimensionless curvature number (C) that weighs the relative importance of in-plane curvature and aperture-induced curvature. Through systematic variation of C, they find in-plane interracial curvature to greatly affect the phase invasion structure. As C is increased from zero, phase invasion fronts transition from highly complicated (IP results) to microscopically smooth. In addition, measurements of fracture phase saturations and entrapped cluster statistics (number, maximum size, structural complication) show differential response between wetting and nonwetting invasion with respect to C that is independent of contact angle hysteresis. Comparison to experimental data available at this time substantiates predicted behavior.

Dispersion of solutes in a variable aperture fracture results from a combination of molecular diffusion and velocity variations in both the plane of the fracture (macrodispersion) and across the fracture aperture (Taylor dispersion). We use a combination of physical experiments and computational simulations to test a theoretical model in which the effective longitudinal dispersion coefficient D(L) is expressed as a sum of the contributions of these three dispersive mechanisms. The combined influence of Taylor dispersion and macrodispersion results in a nonlinear dependence of D(L) on the Peclet number (Pe = V/D(m), where V is the mean solute velocity,is the mean aperture, and D(m) is the molecular diffusion coefficient). Three distinct dispersion regimes become evident: For small Pe (Pe << 1), molecular diffusion dominates resulting in D(L) proportional to Pe0; for intermediate Pe, macrodispersion dominates (D(L) proportional to Pe); and for large Pe, Taylor dispersion dominates (D(L) proportional to Pe2). The Pe range corresponding to these different regimes is controlled by the statistics of the aperture field. In particular, the upper limit of Pe corresponding to the macrodispersion regime increases as the macrodispersivity increases. Physical experiments in an analog, rough-walled fracture confirm the nonlinear Pe dependence of D(L) predicted by the theoretical model. However, the theoretical model underestimates the magnitude of D(L). Computational simulations, using a particle-tracking algorithm that incorporates all three dispersive mechanisms, agree very closely with the theoretical model predictions. The close agreement between the theoretical model and computational simulations is largely because, in both cases, the Reynolds equation describes the flow field in the fracture. The discrepancy between theoretical model predictions and D(L) estimated from the physical experiments appears to be largely, due to deviations from the local cubic law assumed by the Reynolds equation.

The authors designed and conducted experiments in a heterogeneous sand pack where gravity-destabilized nonwetting phase invasion (CO{sub 2} and TCE) could be recorded using high resolution light transmission methods. The heterogeneity structure was designed to be reminiscent of fluvial channel lag cut-and-fill architecture and contain a series of capillary barriers. As invasion progressed, nonwetting phase structure developed a series of fingers and pools; behind the growing front they found nonwetting phase saturation to pulsate in certain regions when viscous forces were low. Through a scale analysis, they derive a series of length scales that describe finger diameter, pool height and width, and regions where pulsation occurs within a heterogeneous porous medium. In all cases, they find that the intrinsic pore scale nature of the invasion process and resulting structure must be incorporated into the analysis to explain experimental results. The authors propose a simple macro-scale structural growth model that assembles length scales for sub-structures to delineate nonwetting phase migration from a source into a heterogeneous domain. For such a model applied at the field scale for DNAPL migration, they expect capillary and gravity forces within the complex subsurface lithology to play the primary roles with viscous forces forming a perturbation on the inviscid phase structure.