Effective parameters for two-phase flow in a porous medium with periodic heterogeneities

Computational simulations of two-phase flow in porous media are used to investigate the feasibility of replacing a porous medium containing heterogeneities with an equivalent homogeneous medium. Simulations are performed for the case of infiltration of a dense nonaqueous phase liquid (DNAPL) in a water-saturated, heterogeneous porous medium. For two specific porous media, with periodic and rather simple heterogeneity patterns, the existence of a representative elementary volume (REV) is studied. Upscaled intrinsic permeabilities and upscaled nonlinear constitutive relationships for two-phase flow systems are numerically calculated and the effects of heterogeneities are evaluated. Upscaled capillary pressure-saturation curves for drainage are found to be distinctly different from the lower-scale curves for individual regions of heterogeneity. Irreducible water saturation for the homogenized medium is found to be much larger than the corresponding lower-scale values. Numerical simulations for both heterogeneous and homogeneous representations of the considered porous media are carried out. Although the homogenized model simulates the spreading behavior of DNAPL reasonably well, it still fails to match completely the results form the heterogeneous simulations. This seems to be due, in part, to the nonlinearities inherent to multiphase flow systems. Although we have focussed on a periodic heterogeneous medium in this study, our methodology is applicable to other forms of heterogeneous media. In particular, the procedure for identification of a REV, and associated upscaled constitutive relations, can be used for randomly heterogeneous or layered media as well.     

A novel two-dimensional model for colloid transport in physically and geochemically heterogeneous porous media

A two-dimensional model for colloid transport in geochemically and physically heterogeneous porous media is presented. The model considers patchwise geochemical heterogeneity, which is suitable to describe the chemical variability of many surficial aquifers with ferric oxyhydroxide-coated porous matrix, as well as spatial variability of hydraulic conductivity, which results in heterogeneous flow field. The model is comprised of a transient fluid flow equation, a transient colloid transport equation, and an equation for the dynamics of colloid deposition and release. Numerical simulations were carried out with the model to investigate the colloid transport behavior in layered and randomly heterogeneous porous media. Results demonstrate that physical and geochemical heterogeneities markedly affect the colloid transport behavior. Layered physical or geochemical heterogeneity can result in distinct preferential flow paths of colloidal particles. Furthermore, the combined effect of layered physical and geochemical heterogeneity may result in enhanced or reduced preferential flow of colloids. Random distribution of physical heterogeneity (hydraulic conductivity) results in a random flow field and an irregularly distributed colloid concentration profile in the porous medium. Contrary to random physical heterogeneity, the effect of random patchwise geochemical heterogeneity on colloid transport behavior is not significant. It is mostly the mean value of geochemical heterogeneity rather than its distribution that governs the colloid transport behavior.     

Modeling solute diffusion in the presence of pore-scale heterogeneity: method development and an application to the Culebra dolomite member of the Rustler Formation, New Mexico, USA

Previous studies have revealed the presence of pore-scale variability in diffusivity in the Culebra (dolomite) member of the Rustler Formation, NM. In this study, eight laboratory-scale diffusion experiments on five Culebra samples were analyzed using a methodology for modeling solute diffusion through porous media in the presence of multiple matrix diffusivities, Dp. A lognormal distribution of Dp is assumed within each of the lab samples. The estimated standard deviation (sigma d) of ln(Dp) within each sample ranges from 0 to 1, with most values lying between 0.5 and 1. The variability over all samples leads to a combined sigma d in the range of 1.0-1.2, which is consistent with the distribution of independently determined formation factor measurements for similar Culebra samples. A comparison of our estimation results to other rock properties suggests that, at the lab-scale, the geometric mean of Dp increases with bulk porosity and the quantity of macroscopic features such as vugs and fractures. However, sigma d appears to be determined by variability within such macroscopic features and/or by micropore-scale heterogeneity. In addition, comparison of these experiments to those at larger spatial scales suggests that increasing sample volume results in an increase in sigma d.     

A hybrid micro-scale model for transport in connected macro-pores in porous media

This paper presents a hybrid model for transport in connected macro-pores in porous media. A pore-scale model is used to parameterize the hybrid model. The hybrid model explicitly models the advection and diffusion of species in the connected macro-pores and treats the porous media around the connected macro-pores as a continuum with effective transport properties. The pore-scale model is used to calculate the effective transport properties of the porous continuum. This approach negates the need to calibrate the hybrid model against experimental data, which is common for continuum-scale models of porous media, and allows an arbitrary microstructure to be considered. The paper presents the multi-scale modeling approach along with the details of the hybrid and pore-scale models. Validation of the model is also presented along with several case studies investigating the applicability of the multi-scale modeling approach to different geometries and transport conditions. The case studies show that the multi-scale modeling approach is accurate for various connected macro-pore geometries given that the porosity of the porous medium around the connected macro-pores is sufficiently small. The accuracy of the hybrid model decreases with increasing porosity of the matrix.     

An assessment of the sensitivity and reliability of the relative reduction factor approach in the development of 8-hr ozone attainment plans

The updated regulatory framework for demonstrating that future 8-hr ozone (O3) design values will be at or below the National Ambient Air Quality Standards (NAAQS) provides guidelines for the development of a State Implementation Plan (SIP) that includes methods based on photochemical modeling and analytical techniques. One of the suggested approaches is the relative reduction factor (RRF) for estimating the efficacy of emission reductions. In this study, the sensitivity of model-predicted responses towards emission reductions to the choice of meteorology and chemical mechanisms was examined. While the different modeling simulations generally were found to be in agreement on whether predicted future-year design values would be above or below the NAAQS for 8-hr O3 at a majority of the monitoring locations in the eastern United States, differences existed for a small percentage of monitors (approximately 6.4%). Another issue investigated was the ability of the attainment demonstration procedure to predict changes in monitored O3 design values. A retrospective analysis was performed by comparing predicted O3 design values from model simulations using emission estimates for 1996 and 2001 with monitored O3 design values for 2001. Results indicated that an average gross error of approximately 5 ppb was present between modeled and observed design values and that, at approximately 27% of all sites, model-predicted and observed design values disagreed as to whether the design value was above or below the NAAQS. Retrospective analyses such as the one presented in this study can provide valuable insights into the strengths and limitations of modeling and analysis techniques used to predict future design values over time periods of a decade or more for the purpose of developing SIPs. Furthermore, such analyses could provide avenues for improvement and added confidence in the use of the RRF approach for addressing attainment of the NAAQS.     

The significance of heterogeneity on mass flux from DNAPL source zones: an experimental investigation

Understanding the process of mass transfer from source zones of aquifers contaminated with organic chemicals in the form of dense non-aqueous phase liquids (DNAPL) is of importance in site management and remediation. A series of intermediate-scale tank experiments was conducted to examine the influence of aquifer heterogeneity on DNAPL mass transfer contributing to dissolved mass emission from source zone into groundwater under natural flow before and after remediation. A Tetrachloroethylene (PCE) spill was performed into six source zone models of increasing heterogeneity, and both the spatial distribution of the dissolution behavior and the net effluent mass flux were examined. Experimentally created initial PCE entrapment architecture resulting from the PCE migration was largely influenced by the coarser sand lenses and the PCE occupied between 30 and 60% of the model aquifer depth. The presence of DNAPL had no apparent effect on the bulk hydraulic conductivity of the porous media. Up to 71% of PCE mass in each of the tested source zone was removed during a series of surfactant flushes, with associated induced PCE mobilization responsible for increasing vertical DNAPL distributions. Effluent mass flux due to water dissolution was also found to increase progressively due to the increase in NAPL-water contact area even though the PCE mass was reduced. Doubling of local groundwater flow velocities showed negligible rate-limited effects at the scale of these experiments. Thus, mass transfer behavior was directly controlled by the morphology of DNAPL within each source zone. Effluent mass flux values were normalized by the up-gradient DNAPL distributions. For the suite of aquifer heterogeneities and all remedial stages, normalized flux values fell within a narrow band with mean of 0.39 and showed insensitivity to average source zone saturations.     

Colloid transport in a geochemically heterogeneous porous medium: aquifer tank experiment and modeling

To examine colloid transport in geochemically heterogeneous porous media at a scale comparable to field experiments, we monitored the migration of silica-coated zirconia colloids in a two-dimensional layered porous media containing sand coated to three different extents by ferric oxyhydroxides. Transport of the colloids was measured over 1.65 m and 95 days. Colloid transport was modeled by an advection-dispersion-deposition equation incorporating geochemical heterogeneity and colloid deposition dynamics (blocking). Geochemical heterogeneity was represented as favorable (ferric oxyhydroxide-coated) and unfavorable (uncoated sand) deposition surface areas. Blocking was modeled as random sequential adsorption (RSA). Release of deposited colloids was negligible. The time to colloid breakthrough after the onset of blocking increased with increasing ferric oxyhydroxide-coated surface area. As the ferric oxyhydroxide surface area increased, the concentration of colloids in the breakthrough decreased. Model-fits to the experimental data were made by inverse solutions to determine the fraction of surface area favorable for deposition and the deposition rate coefficients for the favorable (ferric oxyhydroxide-coated) and unfavorable sites. The favorable deposition rate coefficient was also calculated by colloid filtration theory. The model described the time to colloid breakthrough and the blocking effect reasonably well and estimated the favorable surface area fraction very well for the two layers with more than 1% ferric oxyhydroxide coating. If mica edges in the uncoated sand were considered as favorable surface area in addition to the ferric oxyhydroxide coatings, the model predicted the favorable surface area fraction accurately for the layer with less than 1% ferric oxyhydroxide coating.     

Straining phenomena in bacteria transport through natural porous media

Background, aim, and scope  

Transport of bacteria through natural porous media is an issue of increasing concern arising in several very important environmental processes. These include the percolation of bacteria from fecal waste to drinking water reservoirs, thus leading to a risk for human health, or the bioremediation of contaminated soils in which the bacteria are expected to travel long distances underground in order to reach contaminated areas and degrade chemicals originating from accidental spills. An understanding of bacterial retention and transport mechanisms in porous media would be of great help in the development of models able to predict the distance covered by bacterial suspensions in these situations.     

More general capillary pressure and relative permeability models from fractal geometry

More general capillary pressure and relative permeability models were derived theoretically from fractal modeling of a porous medium. It was found that the new capillary pressure model could be reduced to the frequently-used Brooks–Corey capillary pressure model and the Li–Horne imbibition model when the fractal dimension of a porous medium takes specific values. This also demonstrates that the Brooks–Corey model and the Li-Horne model have a further confirmed theoretical basis. Capillary pressure data measured using mercury intrusion techinque were used to verify the model. The results demonstrated that the new capillary pressure model could represent the capillary pressure curves in those rocks with fracures or with great heterogeneity while the existing models cannot. The new relative permeability models can be reduced to the Brooks–Corey relative permeability model in a specific case. It has been proved theoretically that the relative permeability of each phase in a smooth fracture is only a linear function of its own saturation. Relative permeability data were calculated using the new models and the model results were compared with experimental data measured using a steady-state technique. The comparison demonstrated that the relative permeability models and experimental results were consistent with each other.     

Modeling the Reduction of Vapor Phase Emissions from Surface Soils Due to Soil Matrix Effects: Porosity/Tortuosity Concepts

ABSTRACT

A major route for transport of volatile organic compounds within porous media is vapor phase diffusion. The diffusion rate through a porous medium is less than that through free-air due to the decreased cross-sectional area available for gas movement and the increased path length due to pore tortuosity. Numerous empirical expressions have been published that relate the diffusion coefficient in porous media to the diffusion coefficient in free-air (unobstructed gas phase). Published measurements of relative diffusivity and air-filled porosity were combined into a database. Empirical expressions available in the literature, including the popular Millington-Quirk equation, were evaluated along with a fourth-degree polynomial expression developed by the authors to determine the best type of equation to predict relative diffusivity as a function of air-filled porosity over the domain of values for porosity ranging from 0.071 to 1 for different types of materials. Mean square deviations were used as the statistical test to compare equations. The polynomial expression developed in this project produced a significantly different effective diffusion coefficient (1.3 x 10^-6 m²/sec) compared to values of 9.2 x 10^-6 m²/sec and 3.1 x 10^-6 m²/ sec predicted by forms of the Millington-Quirk equation for a specific case.     

Migration of colloids in discretely fractured porous media: effect of colloidal matrix diffusion

Numerical simulations of colloid transport in discretely fractured porous media were performed to investigate the importance of matrix diffusion of colloids as well as the filtration and remobilization of colloidal particles in both the fractures and porous matrix. To achieve this objective a finite element numerical code entitled COLDIFF was developed. The processes that COLDIFF takes into account include advective-dispersive transport of colloids, filtration and remobilization of colloidal particles in both fractures and porous matrix, and diffusive interactions of colloids between the fractures and porous matrix. Three sets of simulations were conducted to examine the importance of parameters and processes controlling colloid migration. First, a sensitivity analysis was performed using a porous block containing a single fracture to determine the relative importance of various phenomenological coefficients on colloid transport. The primary result of the analysis showed that the porosity of the matrix and the process of colloid filtration in fractures play important roles in controlling colloid migration. Second, simulations were performed to replicate and examine the results of a laboratory column study using a fractured shale saprolite. Results of this analysis showed that the filtration of colloidal particles in the porous matrix can greatly affect the tailing of colloid concentrations after the colloid source was removed. Finally, field-scale simulations were performed to examine the effect of matrix porosity, fracture filtration and fracture remobilization on long-term colloid concentration and migration distance. The field scale simulations indicated that matrix diffusion and fracture filtration can significantly reduce colloid migration distance. Results of all three analyses indicated that in environments where porosity is relatively high and colloidal particles are small enough to diffuse out of fractures, the characteristics of the porous matrix that affect colloid transport become more important than those of the fracture network. Because the properties of the fracture network tend to have greater uncertainty due to difficulties in their measurement relative to those of the porous matrix, prediction uncertainties associated with colloid transport in discretely fractured porous media may be reduced.     

Kinetic modeling of virus transport at the field scale

Bacteriophage removal by soil passage in two field studies was re-analyzed with the goal to investigate differences between one- and two-dimensional modeling approaches, differences between one- and two-site kinetic sorption models, and the role of heterogeneities in the soil properties. The first study involved removal of bacteriophages MS2 and PRDI by dune recharge, while the second study represented removal of MS2 by deep well injection. In both studies, removal was higher during the first meters of soil passage than thereafter. The software packages HYDRUS-ID and HYDRUS-2D, which simulate water flow and solute transport in one- and two-dimensional variably saturated porous media, respectively, were used. The two codes were modified by incorporating reversible adsorption to two types of kinetic sites. Tracer concentrations were used first to calibrate flow and transport parameters of both models before analyzing transport of bacteriophages. The one-dimensional one-site model did not fully describe the tails of the measured breakthrough curves of MS2 and PRD1 from the dune recharge study. While the one-dimensional one-site model predicted a sudden decrease in virus concentrations immediately after the peaks, measured data displayed much smoother decline and tailing. The one-dimensional two-site model simulated the overall behavior of the breakthrough curves very well. The two-dimensional one-site model predicted a more gradual decrease in virus concentrations after the peaks than the one-dimensional one-site model, but not as good as the one-dimensional two-site model. The dimensionality of the problem hence can partly explain the smooth decrease in concentration after peak breakthrough. The two-dimensional two-site model provided the best results. Values for k(att2) and k(det2) could not be determined at the last two of four monitoring wells, thus suggesting that either a second type of kinetic sites is present in the first few meters of dune passage and not beyond the second monitoring well, or that effects of soil heterogeneity and dimensionality of the problem overshadowed this process. Variations between single collector efficiencies were relatively small, whereas collision efficiencies varied greatly. This implies that the nonlinear removal of MS2 and PRD1 is mainly caused by variations in interactions between grain and virus surfaces rather than by physical heterogeneity of the porous medium. Similarly, a two-site model performed better than the one-site model in describing MS2 concentrations for the deep well injection study. However, the concentration data were too sparse in this study to have much confidence in the fitted parameters.     

A pore-scale investigation of a multiphase porous media system

Pore-scale processes govern fundamental behavior in multiphase porous media systems. A high-resolution, three-dimensional image of the interior of a multiphase porous media system was obtained using synchrotron X-ray tomography. The system was imaged at a resolution of 12.46 mum following entrapment of the nonwetting phase at residual saturation. First, the physically representative network structure of the porous media system is extracted from the void space. This provides a direct mapping of the pore bodies and throats and enables pore-level calculations of coordination numbers, aspect ratios, and pore body and throat correlations. Next, algorithms developed to calculate properties of the entrapped nonwetting phase, such as volume, sphericity, interfacial area, and orientation, are applied to the residual nonwetting phase blobs. Finally, correlations between the pore network structure and nonwetting phase characteristics are examined. As expected, it was found that the nonwetting phase was trapped primarily in the largest pore spaces, the pore bodies with the highest aspect ratios, and the pore bodies with the highest coordination numbers. This work shows that, while there may be limitations related to the ability to capture REV-sized domains for some of the multiphase flow properties and phenomena, high-resolution X-ray tomography is able to provide the high quality datasets needed to observe and quantify the pore-scale phenomena and processes that govern multiphase flow in unconsolidated porous media systems.     

Release of colloidal particles in natural porous media by monovalent and divalent cations

We study mobilization of colloidal particles from natural porous media, such as soils and groundwater aquifers. Extensive laboratory scale column experiments of particle release from four different subsurface materials are presented. The important characteristics of the release process are (i) its non-exponential kinetics, (ii) the finite supply of colloidal particles and (iii) the strong dependence of the release kinetic on the nature of the adsorbed cations. Particle release depends most sensitively on the relative saturation of the medium with divalent cations. We propose a mathematic model, which captures all these aspects quantitatively, and can be used to describe the coupling between transport of major cations and the release of colloidal particles. The present experimental investigations as well as the developed modeling framework represent an important step towards the understanding of colloid-facilitated transport phenomena in natural porous media.     

Quantitative imaging of contaminant distributions in heterogeneous porous media laboratory experiments

Intermediate-scale laboratory experiments on heterogeneous porous media have been increasingly used for the study of saturated and unsaturated ground water systems. While the ability to reproduce field-scale heterogeneity in these experiments has advanced, the use of visualization or image analysis methods to characterize the spatial distribution of solute concentrations has largely remained at the homogeneous media level. To advance these imaging techniques we developed a generic image analysis package that, for the first time, automatically segments regions in photographic images that require unique concentration calibration curves due to varying porous media properties or lighting nonuniformities. As a robust test, our image analysis package was applied to an intermediate-scale flow tank experiment characterized by a correlated random permeability field with unprecedented resolution. Twenty-five distinct classes of porous media were developed and binned to the synthetic permeability field, creating an experimental field of 3456 rectangular cells and thereby ensuring the emplaced field closely matched the statistics of the original continuous distribution. Concentration distributions were determined for an experimental tracer run and the corresponding dispersion parameters were calculated. The closeness of the experimental, image-processed longitudinal dispersivity (4.6 x 10(-2) m) to that obtained from the field statistics (9.1 x 10(-2) m) verifies our image analysis technique.     

Meteorological factors of ozone predictability at Houston, Texas

Several ozone modeling approaches were investigated to determine if uncertainties in the meteorological data would be sufficiently large to limit the application of physically realistic ozone (O3) forecast models. Three diagnostic schemes were evaluated for the period of May through September 1997 for Houston, TX. Correlations between measured daily maximum and model calculated O3 air concentrations were found to be 0.70 using a linear regression model, 0.65 using a non-advective box model, and 0.49 using a three-dimensional (3-D) transport and dispersion model. Although the regression model had the highest correlation, it showed substantial underestimates of the highest concentrations. The box model results were the most similar to the regression model and did not show as much underestimation. The more complex 3-D modeling approach yielded the worst results, likely resulting from O3 maxima that were driven by local factors rather than by the transport of pollutants from outside of the Houston domain. The highest O3 concentrations at Houston were associated with light winds and meandering trajectories. A comparison of the gridded meteorological data used by the 3-D model to the observations showed that the wind direction and speed values at Houston differed most on those days on which the O3 underestimations were the greatest. These periods also tended to correspond with poor precipitation and temperature estimates. It is concluded that better results are not just obtained through additional modeling complexity, but there needs to be a comparable increase in the accuracy of the meteorological data.     

Analysis of field observations of tracer transport in a fractured till

We analyze a set of observations from a recently published, field-scale tracer test in a fractured till. These observations demonstrate a dominant, underlying non-Fickian behavior, which cannot be quantified using traditional modeling approaches. We use a continuous time random walk (CTRW) approach which thoroughly accounts for the measurements, and which is based on a physical picture of contaminant motion that is consistent with the geometric and hydraulic characterization of the fractured formation. We also incorporate convolution techniques in the CTRW theory, to consider transport between different regions containing distinct heterogeneity patterns. These results enhance the possibility that limitations in predicting non-Fickian modes of contaminant migration can be overcome.     

An overview of uncertainties in modelling groundwater solute transport

Model predictions are uncertain because of uncertainties on future and/or anthropogenic stresses, parameter values and conceptual models. The first two groups of problems can be addressed through rather systematic methods (scenario analysis, error transmission techniques, automatic calibration algorithms, etc.). However, conceptual uncertainties are rarely given adequate attention. The objective of this paper is to synthesize conceptual difficulties associated with transport. These include: (1) processes that are significant at small scales may not be relevant at large scales; (2) inversely, new processes (e.g., dispersion) emerge in response to increase in scale and the way to represent them may depend on the assumed model structure; (3) the observed shapes of both breakthrough curves and pollutant plumes are not well represented by the classical transport equation; (4) porosities and dispersivities derived from field tracer tests often exhibit a directional dependence; etc. Though not directly related to solute transport, scale effects on hydraulic conductivity may also affect solute transport modelling. When these anomalies are examined, it is concluded that they are directly or indirectly caused by heterogeneity. Current approaches for dealing with heterogeneity can be divided into stochastic and deterministic. Stochastic methods have been successful in explaining qualitatively some anomalies of solute transport, but they appear to be far from reaching a stage at which they can be used routinely for solving realistic field problems. On the other hand, when applied with care, deterministic methods have been successfully used in actual problems. Yet, it can be argued that they fail to account for small-scale variability of concentrations so that they would become ineffective when dealing with nonlinear processes, such as chemical reactions. Relevance of on-going research for overcoming these difficulties is discussed.     

Assessing and managing freshwater ecosystems vulnerable to environmental change

Freshwater ecosystems are important for global biodiversity and provide essential ecosystem services. There is consensus in the scientific literature that freshwater ecosystems are vulnerable to the impacts of environmental change, which may trigger irreversible regime shifts upon which biodiversity and ecosystem services may be lost. There are profound uncertainties regarding the management and assessment of the vulnerability of freshwater ecosystems to environmental change. Quantitative approaches are needed to reduce this uncertainty. We describe available statistical and modeling approaches along with case studies that demonstrate how resilience theory can be applied to aid decision-making in natural resources management. We highlight especially how long-term monitoring efforts combined with ecological theory can provide a novel nexus between ecological impact assessment and management, and the quantification of systemic vulnerability and thus the resilience of ecosystems to environmental change.     

The velocity of DNAPL fingering in water-saturated porous media: laboratory experiments and a mobile-immobile-zone model.

Dense nonaqueous phase liquids (DNAPLs) are immiscible with water and can give rise to highly fingered fluid distributions when infiltrating through water-saturated porous media. In this paper, a conceptual mobile-immobile-zone (MIZ) model is presented to describe the structure of a DNAPL finger in water-saturated porous media and the velocity of finger propagation. A finger is composed of a finger body and a tip. The finger body has a mobile core and an immobile sheath. All the DNAPL within the tip of a finger is mobile. Lab experiments utilizing image analyses of a DNAPL (PCE) penetrating into water-saturated homogeneous glass beads were carried out in a two-dimensional transparent chamber. The results show that the fingers elongated almost linearly with time. The fingers did not grow laterally after the tip of the finger had passed. The average finger diameters were between 3.9 and 5.4 mm for PCE propagation in water-saturated glass bead porous media with mean particle diameters from 0.32 to 1.36 mm. The estimated mobile core diameters were 51-60% of the average finger diameters.