An application of the theory of kinematics of mixing to the study of tropospheric dispersion

The most common technique used for numerical simulations of tracer mixing is that of the numerical solution of the advection–diffusion equation with the unresolved fluxes parameterized using the similarity theory. Despite correct predictions of the overall directions of transport, models based on a numerical solution of the advection–diffusion equation lack sufficient accuracy to correctly reproduce the coupling of mixing with small scale processes which are sensitive to the microstructure of the tracer distribution. The objective of this paper is to revisit the basic formalism employed in numerical models used to investigate atmospheric tracers. The main mathematical method proposed here is the theory of kinematics of mixing which could be applied effectively for simulations of atmospheric transport processes. At the beginning of the paper, we introduce simple mathematical transformations in order to demonstrate how complex topological structures are created by mixing processes. These idealistic flow systems are essential to explain transport properties of much more complex three-dimensional geophysical flows. An example of the application of the kinematics of mixing to the analysis of tracer transport on a planetary scale is presented in the following sections. The complex filamentary structures simulated in the numerical experiment are evaluated using some commonly applied statistical measures in order to compare the results with the data published in the literature. The results of the experiment are also analysed with the help of simple conceptual models of fluid filaments. The microstructure of the tracer distribution introduced in the paper is essential to increase our understanding of atmospheric transport and to develop more realistic parameterizations of small-scale mixing. The presented results could also be used to improve calculations of the coupling between microphysical processes and tracer mixing.     

Modelling of nonreactive tracer dipole tests in a shear zone at the Grimsel test site

The investigation of the migration of a high pH plume in a fractured shear zone is foreseen by a long-term experiment at the Grimsel rock laboratory. In order to characterise the initial conditions for the long-term experiment and to evaluate an optimal hydraulic in situ set-up, several dipole experiments with nonreacting tracers have been performed. The dipole experiments differ in geometry, pumping rates and orientation to the background water flow. Several single and double-porosity models have been applied to fit the results of these dipole tracer tests in order to extract values for some transport parameters and discriminate for certain transport processes. A two-dimensional porous medium approach was successfully used to fit tracer breakthrough curves measured for a dipole experiment. A model based on a one-dimensional dual porous medium approach was also successful, although the applied hydraulic dipole, with similar injection and extraction rates, suggests the existence of an extended two-dimensional flow field. For the two-dimensional porous medium approach, tracer breakthrough could only be fitted with a complex flow field geometry within the heterogeneous fractured shear zone. The heterogeneity was generated by heterogeneous porosity and hydraulic permeability distributions. Predictions for further dipole geometries and a sorbing tracer have been calculated by means of both models using the flow and transport parameters deduced from fits for a single dipole experiment. This allows for comparison with the measured breakthrough of sorbing tracers. The foreseen experiment with sorbing (radionuclide) tracers will help decide on the appropriate approach that should be used to describe such dipole experiments in this shear zone. Additionally, the migration and spreading of a solution with high pH has been calculated taking into account mineral dissolution and precipitation in a two-dimensional porous medium approach in order to estimate the amount and character of the mineral reactions induced by the interaction between the high pH solution and the rock.     

Fluorescent dye imaging of the volume sampled by single well forced-gradient tracer tests evaluated in a laboratory-scale aquifer physical model

This study presents a new method to visualise forced-gradient tracer tests in 2-D using a laboratory-scale aquifer physical model. Experiments were designed to investigate the volume of aquifer sampled in vertical dipole flow tracer tests (DFTT) and push-pull tests (PPT), using a miniature monitoring well and straddle packer arrangement equipped with solute injection and recovery chambers. These tests have previously been used to estimate bulk aquifer hydraulic and transport properties for the evaluation of natural attenuation and other remediation approaches. Experiments were performed in a silica glass bead-filled box, using a fluorescent tracer (fluorescein) to deduce conservative solute transport paths. Digital images of fluorescein transport were captured under ultraviolet light and processed to analyse tracer plume geometry and obtain point-concentration breakthrough histories. Inorganic anion mixtures were also used to obtain conventional tracer breakthrough histories. Concentration data from the conservative tracer breakthrough curves was compared with the digital images and a well characterised numerical model. The results show that the peak tracer breakthrough response in dipole flow tracer tests samples a zone of aquifer close to the well screen, while the sampling volume of push-pull tests is limited by the length of the straddle packers used. The effective sampling volume of these single well forced-gradient tests in isotropic conditions can be estimated with simple equations. The experimental approach offers the opportunity to evaluate under controlled conditions the theoretical basis, design and performance of DFTTs and PPTs in porous media in relation to measured flow and transport properties.     

A combined first-order closure/gaussian dispersion model

A dispersion model incorporating a large wind direction shear has been developed for an industrialized coastal area with an indented coastline and rough terrain. The model is based on a first-order closure solution for the vertical spread and a Gaussian profile for the lateral spread. Lateral spread is obtained by numerical integration of turbulent energy spectra. The model is verified against a large number of tracer experiments with near surface release within industrial complexes. The model is shown to correctly simulate shear-enhanced spread out to a distance of 10 km from the source. Part of the observed scatter is believed to be the result of an oversimplified plume rise calculation.     

Simulations of flow around a cubical building: comparison with towing-tank data and assessment of radiatively induced thermal effects

A three-dimensional (3-D) computational fluid dynamics (CFD) model, coupled with a meteorological radiation and surface physics package, is used to model the mean flow field and tracer dispersion in the vicinity of an idealized cubical building. We first compare the simulations with earlier numerical studies as well as towing-tank laboratory experiments, where radiation effects were not included. Our simulations capture most of the features revealed by the towing-tank data, including the variation of the flow reattachment point as a function of Froude number and the induction of a prominent lee wave in the low Froude number regime. The simulated tracer concentration also compares very favorably with the data.We then assess the thermal effects due to radiative heating on the ground and building including shading by the building, on the mean flow and tracer dispersion. Our simulations show that convergence within and beyond the cavity zone causes a substantial lofting of the air mass downstream from the building. This lofting results from the combination of thermal heating of the ground and building roof, and vortex circulation associated with the horseshoe eddy along the lateral sides of the building. The specific effect of shading on the flow field is isolated by comparing simulations for which the radiative heating and shading patterns are kept constant, but the environmental wind direction is altered. It is found that the shading exerts local cooling, which can be combined into the overall thermodynamic interaction, described above, to effectively alter the circulation downstream from the building.     

Kinematic wave approximation to solute transport along preferred flow paths in soils

A routing procedure is introduced which accounts for the loss of a conservative solute tracer from preferred paths during macropore flow. Water flow is treated as a series of kinematic waves from which the tracer is lost due to mixing previously stored soil water, and an expression for solute loss is added to a previously developed model. The model parameters are estimated through experiments at three different input rates applied to a column of a macroporous forest soil.The results of seven experimental runs indicate that solute losses are consistently highest at the early stages of infiltration and drainage flow. An empirical relationship is proposed which links the frequency distribution of the flow parameter with that for solute loss from the preferred path during transient water flow and solute transport.     

Transport and degradation of propyleneglycol and potassium acetate in the unsaturated zone

De-icing chemicals used during the winter season are potential pollutants for the groundwater underneath the new main airport of Norway. Several field experiments examining the transport and degradation of propyleneglycol (PG), potassium acetate (KAc) and non-reactive tracers were performed in a lysimeter trench under natural snowmelting conditions. Chemicals were applied underneath the snow cover and the transport in a heterogeneous coarse sandy soil was examined by extracting soil water from 30 or 40 suction cups placed at five depths between 0.4 and 2.4 m depth. Transport and degradation was analysed by spatial moment calculations. The de-icing chemicals showed the same basic displacement as chemically inactive tracers, an initial fast transport during the melting period followed by a period of stagnation throughout the summer season. PG seemed to be displaced to greater depths compared to non-reactive tracer after the first application. However, computer simulations of transport and degradation in a heterogeneous unsaturated soil showed that decreasing degradation constants with depth can generate a downward movement of the centre of mass without any flow occurring in the system. Potassium acetate showed some adsorption, with calculated retardation factors of approximately 1.3 and 1.2. The degradation rate constant for PG was calculated to be 0.015 day-1 in 1994 and increased to 0.047 day-1 in the second application made in 1995. The degradation rate constant for acetate was estimated to be 0.02 day-1. Increased manganese concentrations seem to be a good indicator of degradation of PG and Ac.     

First-passage-time transfer functions for groundwater tracer tests conducted in radially convergent flow

Forced-gradient groundwater tracer tests may be conducted using a variety of hydraulic schemes, so it is useful to have simple semi-analytic models available that can examine various injection/withdrawal scenarios. Models for radially convergent tracer tests are formulated here as transfer functions, which allow complex tracer test designs to be simulated by a series of simple mathematical expressions. These mathematical expressions are given in Laplace space, so that transfer functions may be placed in series by simple multiplication. Predicted breakthrough is found by numerically inverting the composite transfer function to the time-domain, using traditional computer programs or commercial mathematical software. Transport is assumed to be dictated by a radially convergent or uniform flow field, and is based upon an exact first-passage-time solution of the backward Fokker–Planck equation. These methods are demonstrated by simulating a weak-dipole tracer test conducted in a fractured granite formation, where mixing in the injection borehole is non-ideal.     

Modelling tritium and phosphorus transport by preferential flow in structured soil

Subsurface solute transport through structured soil is studied by model interpretation of experimental breakthrough curves from tritium and phosphorus tracer tests in three intact soil monoliths. Similar geochemical conditions, with nearly neutral pH, were maintained in all the experiments. Observed transport differences for the same tracer are thus mainly due to differences in the physical transport process between the different monoliths. The modelling is based on a probabilistic Lagrangian approach that decouples physical and chemical mass transfer and transformation processes from pure and stochastic advection. Thereby, it enables explicit quantification of the physical transport process through preferential flow paths, honouring all independently available experimental information. Modelling of the tritium breakthrough curves yields a probability density function of non-reactive solute travel time that is coupled with a reaction model for linear, non-equilibrium sorption–desorption to describe the phosphorus transport. The tritium model results indicate that significant preferential flow occurs in all the experimental soil monoliths, ranging from 60–100% of the total water flow moving through only 25–40% of the total water content. In agreement with the fact that geochemical conditions were similar in all experiments, phosphorus model results yield consistent first-order kinetic parameter values for the sorption–desorption process in two of the three soil monoliths; phosphorus transport through the third monolith cannot be modelled because the apparent mean transport rate of phosphorus is anomalously rapid relative to the non-adsorptive tritium transport. The occurrence of preferential flow alters the whole shape of the phosphorus breakthrough curve, not least the peak mass flux and concentration values, and increases the transported phosphorus mass by 2–3 times relative to the estimated mass transport without preferential flow in the two modelled monoliths.     

The effects of matrix diffusion on radionuclide migration in rock column experiments

Rock column experiments were performed to examine the effects of matrix diffusion and hydrodynamic dispersion on the migration of radionuclides at the laboratory scale. Tritiated water and chloride transportation was studied in intact mica gneiss and in altered more porous tonalite columns with narrow flow channels. The column diffusion properties were estimated prior to water flow experiments using the gas diffusion method with helium as the tracer gas. The numerical compartment model for advection and dispersion, with and without matrix diffusion, was used to interpret the tracer transport in the columns. Matrix diffusion behavior was also distinguished from dominating hydrodynamic dispersion in rock column experiments at the slowest water flow rates.     

An interpretation of potential scale dependence of the effective matrix diffusion coefficient

Matrix diffusion is an important process for solute transport in fractured rock, and the matrix diffusion coefficient is a key parameter for describing this process. Previous studies have indicated that the effective matrix diffusion coefficient values, obtained from a large number of field tracer tests, are enhanced in comparison with local values and may increase with test scale. In this study, we have performed numerical experiments to investigate potential mechanisms behind possible scale-dependent behavior. The focus of the experiments is on solute transport in flow paths having geometries consistent with percolation theories and characterized by multiple local flow loops formed mainly by small-scale fractures. The water velocity distribution through a flow path was determined using discrete fracture network flow simulations, and solute transport was calculated using a previously derived impulse-response function and a particle-tracking scheme. Values for effective (or up-scaled) transport parameters were obtained by matching breakthrough curves from numerical experiments with an analytical solution for solute transport along a single fracture. Results indicate that a combination of local flow loops and the associated matrix diffusion process, together with scaling properties in flow path geometry, seems to be an important mechanism causing the observed scale dependence of the effective matrix diffusion coefficient (at a range of scales).     

混凝搅拌内部流场数值模拟分析

通过对圆形搅拌容器内部流场的数值模拟,从湍流速度场、湍动耗散率和湍动能3个角度来分析该容器内的二维空间流态变化,及其对混凝过程中颗粒碰撞和絮体成长的影响,针对其不足之处,提出改进模型:方形、圆形挡板、多边形、圆角和方形导流板模型,并分别比较各模型在不同转速下的内部流场的变化。结果表明,在相同截面积的条件下,圆角模型更有利于颗粒的碰撞粘结和絮体的成长。     

Changes in physical properties of a compost biofilter treating hydrogen sulfide

A technique is presented that can be used to estimate the changes in physical structure in a natural biofilter packing medium, such as compost, over time. The technique applies information from tracer studies, grain size distribution, and pressure drop analysis to a model that estimates the number of channels, average channel diameter, number of particles, and specific surface area of the medium. Important operational factors, such as moisture content, pressure drop, and sulfate accumulation also were evaluated both in a conventionally operated biofilter and in one operated with periodic compost mixing. In the conventionally operated laboratory-scale compost biofilter, hydrogen sulfide (H2S) removal efficiency decreased from 100% to approximately 90% over 206 days of operation. In a similar system, operated with compost mixing, the H2S removal efficiency was maintained near 100%. Variations in media moisture conditions and specific surface area can explain the results observed in this study. Under conventional operation, drying near the inlet disintegrated the compost particles, producing a large number of particles and flow channels and increasing the specific surface area. At the top of the column, where moisture was added, particle size increased and specific surface area decreased. In the column with media mixing, moisture content, particle size, and specific surface area remained homogeneous.     

An axisymmetric diffusion experiment for the determination of diffusion and sorption coefficients of rock samples

Diffusion anisotropy is a critical property in predicting migration of substances in sedimentary formations with very low permeability. The diffusion anisotropy of sedimentary rocks has been evaluated mainly from laboratory diffusion experiments, in which the directional diffusivities are separately estimated by through-diffusion experiments using different rock samples, or concurrently by in-diffusion experiments in which only the tracer profile in a rock block is measured. To estimate the diffusion anisotropy from a single rock sample, this study proposes an axisymmetric diffusion test, in which tracer diffuses between a cylindrical rock sample and a surrounding solution reservoir. The tracer diffusion between the sample and reservoir can be monitored from the reservoir tracer concentrations, and the tracer profile could also be obtained after dismantling the sample. Semi-analytical solutions are derived for tracer concentrations in both the reservoir and sample, accounting for an anisotropic diffusion tensor of rank two as well as the dilution effects from sampling and replacement of reservoir solution. The transient and steady-state analyses were examined experimentally and numerically for different experimental configurations, but without the need for tracer profiling. These experimental configurations are tested for in- and out-diffusion experiments using Koetoi and Wakkanai mudstones and Shirahama sandstone, and are scrutinized by a numerical approach to identify favorable conditions for parameter estimation. The analysis reveals the difficulty in estimating diffusion anisotropy; test configurations are proposed for enhanced identifiability of diffusion anisotropy. Moreover, it is demonstrated that the axisymmetric diffusion test is efficient in obtaining the sorption parameter from both steady-state and transient data, and in determining the effective diffusion coefficient if isotropic diffusion is assumed. Moreover, measuring reservoir concentrations in an axisymmetric diffusion experiment coupled with tracer profiling may be a promising approach to estimate of diffusion anisotropy of sedimentary rocks.     

Variable-density flow in heterogeneous porous media — Laboratory experiments and numerical simulations

Konz, M., Ackerer, P., Younes, A., Huggenberger, P., Zechner, E., 2009a. 2D Stable Layered Laboratory-scale Experiments for Testing Density-coupled Flow Models. Water Resources Research, 45. doi:10.1029/2008WR007118., a series of laboratory-scale 2D tank experiments were conducted and accurately simulated for density driven flow problems on homogeneous porous media. In the present work, we extended the numerical and experimental studies to heterogeneous problems. The heterogeneous porous medium was constructed with a low permeability zone in the centre of the tank and had well-defined parameters and boundary conditions. Concentration distributions were measured in high resolution using a photometric method and an image analysis technique. The numerical model used for the simulations was based on efficient advanced approximations for both spatial and temporal discretizations. The Method Of Lines (MOL) was used to allow higher-order temporal discretization. Three different boundary conditions, corresponding to different localizations of the inflow and the outflow openings at the opposite edges of the tank, were applied to investigate different flow scenarios in the heterogeneous porous medium flow tank. Simulation results of all three density coupled experiments revealed a density-dependent behavior of dispersion. Thus, a reduction of dispersivites was required to obtain a good matching of the experimental data. The high quality of the experiments enabled a detailed testing of numerical variable-density flow codes under heterogeneous conditions. Therefore, the experiments were considered to be reliable benchmark tests.     

Evaluating the ability of a numerical weather prediction model to forecast tracer concentrations during ETEX 2

In this paper the meteorological processes responsible for transporting tracer during the second ETEX (European Tracer EXperiment) release are determined using the UK Met Office Unified Model (UM). The UM predicted distribution of tracer is also compared with observations from the ETEX campaign. The dominant meteorological process is a warm conveyor belt which transports large amounts of tracer away from the surface up to a height of 4 km over a 36 h period. Convection is also an important process, transporting tracer to heights of up to 8 km. Potential sources of error when using an operational numerical weather prediction model to forecast air quality are also investigated. These potential sources of error include model dynamics, model resolution and model physics. In the UM a semi-Lagrangian monotonic advection scheme is used with cubic polynomial interpolation. This can predict unrealistic negative values of tracer which are subsequently set to zero, and hence results in an overprediction of tracer concentrations. In order to conserve mass in the UM tracer simulations it was necessary to include a flux corrected transport method. Model resolution can also affect the accuracy of predicted tracer distributions. Low resolution simulations (50 km grid length) were unable to resolve a change in wind direction observed during ETEX 2, this led to an error in the transport direction and hence an error in tracer distribution. High resolution simulations (12 km grid length) captured the change in wind direction and hence produced a tracer distribution that compared better with the observations. The representation of convective mixing was found to have a large effect on the vertical transport of tracer. Turning off the convective mixing parameterisation in the UM significantly reduced the vertical transport of tracer. Finally, air quality forecasts were found to be sensitive to the timing of synoptic scale features. Errors in the position of the cold front relative to the tracer release location of only 1 h resulted in changes in the predicted tracer concentrations that were of the same order of magnitude as the absolute tracer concentrations.     

3-D numerical evaluation of density effects on tracer tests

In this paper we present numerical simulations carried out to assess the importance of density-dependent flow on tracer plume development. The scenario considered in the study is characterized by a short-term tracer injection phase into a fully penetrating well and a natural hydraulic gradient. The scenario is thought to be typical for tracer tests conducted in the field. Using a reference case as a starting point, different model parameters were changed in order to determine their importance to density effects. The study is based on a three-dimensional model domain. Results were interpreted using concentration contours and a first moment analysis. Tracer injections of 0.036 kg per meter of saturated aquifer thickness do not cause significant density effects assuming hydraulic gradients of at least 0.1%. Higher tracer input masses, as used for geoelectrical investigations, may lead to buoyancy-induced flow in the early phase of a tracer test which in turn impacts further plume development. This also holds true for shallow aquifers. Results of simulations with different tracer injection rates and durations imply that the tracer input scenario has a negligible effect on density flow. Employing model cases with different realizations of a log conductivity random field, it could be shown that small variations of hydraulic conductivity in the vicinity of the tracer injection well have a major control on the local tracer distribution but do not mask effects of buoyancy-induced flow.     

Numerical calculation of flow and stack-gas concentration fluctuation around a cubical building

A numerical simulation model was developed to predict the instantaneous concentration fluctuation of a plume and applied to stack-gas diffusion around a cubical building. The flow field, including an instantaneous velocity component, was predicted using the large eddy simulation (LES) method in the developed numerical model. Then, the instantaneous concentration fluctuation was predicted using the obtained unsteady flow field. Concentration was calculated using the finite difference method, in which the LES is expanded for concentration, and the puff method, in which small volumes of the tracer gas are divided and combined according to the calculation mesh sizes. In order to avoid numerical viscous effects, a puff method and finite difference method were applied separately in the regions near and far from the stack-gas release point, respectively. Then, the flow field around a cubical building and the diffusion of stack-gas, emitted from an elevated point source at an upstream position of the building, were calculated using the model mentioned above. Numerical calculation results were compared with those obtained in wind tunnel experiments in which concentration fluctuation was measured using high-response flame ionization detectors. Although there were some discrepancies in the flow field between the calculated results and those of wind tunnel experiments, e.g., the calculated windward length of a cavity region behind the building, the calculated mean velocity and turbulent intensity showed good agreement with those of the wind tunnel experiments. Furthermore, the calculated concentration fluctuation showed good agreement with that in the wind tunnel, not only regarding the features of fluctuating concentration signals, but also statistic quantities, viz., mean concentration, fluctuation intensity and high-concentration values.     

3-D numerical evaluation of density effects on tracer tests

In this paper we present numerical simulations carried out to assess the importance of density-dependent flow on tracer plume development. The scenario considered in the study is characterized by a short-term tracer injection phase into a fully penetrating well and a natural hydraulic gradient. The scenario is thought to be typical for tracer tests conducted in the field. Using a reference case as a starting point, different model parameters were changed in order to determine their importance to density effects. The study is based on a three-dimensional model domain. Results were interpreted using concentration contours and a first moment analysis. Tracer injections of 0.036 kg per meter of saturated aquifer thickness do not cause significant density effects assuming hydraulic gradients of at least 0.1%. Higher tracer input masses, as used for geoelectrical investigations, may lead to buoyancy-induced flow in the early phase of a tracer test which in turn impacts further plume development. This also holds true for shallow aquifers. Results of simulations with different tracer injection rates and durations imply that the tracer input scenario has a negligible effect on density flow. Employing model cases with different realizations of a log conductivity random field, it could be shown that small variations of hydraulic conductivity in the vicinity of the tracer injection well have a major control on the local tracer distribution but do not mask effects of buoyancy-induced flow.