Modeling of non-reactive solute transport in fractured clayey till during variable flow rate and time

Fractures and biopores can act as preferential flow paths in clay aquitards and may rapidly transmit contaminants into underlying aquifers. Reliable numerical models for assessment of groundwater contamination from such aquitards are needed for planning, regulatory and remediation purposes. In this investigation, high resolution preferential water-saturated flow and bromide transport data were used to evaluate the suitability of equivalent porous medium (EPM), dual porosity (DP) and discrete fracture/matrix diffusion (DFMD) numerical modeling approaches for assessment of flow and non-reactive solute transport in clayey till. The experimental data were obtained from four large undisturbed soil columns (taken from 1.5 to 3.5 m depth) in which biopores and channels along fractures controlled 96-99% of water-saturated flow. Simulating the transport data with the EPM effective porosity model (FRACTRAN in EPM mode) was not successful because calibrated effective porosity for the same column had to be varied up to 1 order of magnitude in order to simulate solute breakthrough for the applied flow rates between 11 and 49 mm/day. Attempts to simulate the same data with the DP models CXTFIT and MODFLOW/MT3D were also unsuccessful because fitted values for dispersion, mobile zone porosity, and mass transfer coefficient between mobile and immobile zones varied several orders of magnitude for the different flow rates, and because dispersion values were furthermore not physically realistic. Only the DFMD modeling approach (FRACTRAN in DFMD mode) was capable to simulate the observed changes in solute transport behavior during alternating flow rate without changing values of calibrated fracture spacing and fracture aperture to represent the macropores.     

Interpretation of injection-withdrawal tracer experiments conducted between two wells in a large single fracture

Tracer experiments conducted using a flow field established by injecting water into one borehole and withdrawing water from another are often used to establish connections and investigate dispersion in fractured rock. As a result of uncertainty in the uniqueness of existing models used for interpretation, this method has not been widely used to investigate more general transport processes including matrix diffusion or advective solute exchange between mobile and immobile zones of fluid. To explore the utility of the injection-withdrawal method as a general investigative tool and with the intent to resolve the transport processes in a discrete fracture, two tracer experiments were conducted using the injection-withdrawal configuration. The experiments were conducted in a fracture which has a large aperture (>500 microm) and horizontally pervades a dolostone formation. One experiment was conducted in the direction of the hydraulic gradient and the other in the direction opposite to the natural gradient. Two tracers having significantly different values of the free-water diffusion coefficient were used. To interpret the experiments, a hybrid numerical-analytical model was developed which accounts for the arcuate shape of the flow field, advection-dispersion in the fracture, diffusion into the matrix adjacent to the fracture, and the presence of natural flow in the fracture. The model was verified by comparison to a fully analytical solution and to a well-known finite-element model. Interpretation of the tracer experiments showed that when only one tracer, advection-dispersion, and matrix diffusion are considered, non-unique results were obtained. However, by using multiple tracers and by accounting for the presence of natural flow in the fracture, unique interpretations were obtained in which a single value of matrix porosity was estimated from the results of both experiments. The estimate of porosity agrees well with independent measurements of porosity obtained from core samples. This suggests that: (i) the injection-withdrawal method is a viable tool for the investigation of general transport processes provided all relevant experimental conditions are considered and multiple conservative tracers are used; and (ii) for the conditions of the experiments conducted in this study, the dominant mechanism for exchange of solute between the fracture and surrounding medium is matrix diffusion.     

Two-phase flow and transport in a single fracture-porous medium system

The prognosis for the remediation of contaminated fractured media is much worse than that for more homogeneous units. Fractures act as conduits for the flow of dense non-aqueous phase liquids (DNAPLs), while diffusion is responsible for the storage of dissolved mass in the surrounding matrix. A numerical model incorporating aqueous phase transport in a variable-aperture fracture and its surrounding matrix is developed and coupled with an existing two-phase flow model. The processes of transient two-phase flow, non-equilibrium dissolution, advective–dispersive transport in the fracture, and three-dimensional matrix diffusion are included in the model. Results from various investigations show that the DNAPL distribution is very sensitive to variations in aperture within a single fracture. Diffusion-controlled mass removal from both the matrix and from the hydraulically inaccessible zones within the fracture itself result in extremely large time frames for significant mass removal from these systems. Success in aqueous phase mass removal from the matrix is very sensitive to the effective fracture spacing. The hydraulic gradient in the fracture only affects the amount of water removed from the system, and does not greatly affect the amount of time required to remove the contaminant mass from the matrix. The ability to remove mass is somewhat sensitive to the porosity and effective matrix diffusion coefficient over the range of expected values.     

Field-scale effective matrix diffusion coefficient for fractured rock: results from literature survey

Matrix diffusion is an important mechanism for solute transport in fractured rock. We recently conducted a literature survey on the effective matrix diffusion coefficient, D_m^e, a key parameter for describing matrix diffusion processes at the field scale. Forty field tracer tests at 15 fractured geologic sites were surveyed and selected for the study, based on data availability and quality. Field-scale D_m^e values were calculated, either directly using data reported in the literature, or by reanalyzing the corresponding field tracer tests. The reanalysis was conducted for the selected tracer tests using analytic or semi-analytic solutions for tracer transport in linear, radial, or interwell flow fields. Surveyed data show that the scale factor of the effective matrix diffusion coefficient (defined as the ratio of D_m^e to the lab-scale matrix diffusion coefficient, D_m, of the same tracer) is generally larger than one, indicating that the effective matrix diffusion coefficient in the field is comparatively larger than the matrix diffusion coefficient at the rock-core scale. This larger value can be attributed to the many mass-transfer processes at different scales in naturally heterogeneous, fractured rock systems.Furthermore, we observed a moderate, on average trend toward systematic increase in the scale factor with observation scale. This trend suggests that the effective matrix diffusion coefficient is likely to be statistically scale-dependent. The scale-factor value ranges from 0.5 to 884 for observation scales from 5 to 2000 m. At a given scale, the scale factor varies by two orders of magnitude, reflecting the influence of differing degrees of fractured rock heterogeneity at different geologic sites. In addition, the surveyed data indicate that field-scale longitudinal dispersivity generally increases with observation scale, which is consistent with previous studies. The scale-dependent field-scale matrix diffusion coefficient (and dispersivity) may have significant implications for assessing long-term, large-scale radionuclide and contaminant transport events in fractured rock, both for nuclear waste disposal and contaminant remediation.     

Simulation and analysis of solute transport in 2D fracture/pipe networks: the SOLFRAC program

The Time Domain Random Walk (TDRW) method has been recently developed by Delay and Bodin [Delay, F. and Bodin, J., 2001. Time domain random walk method to simulate transport by advection-dispersion and matrix diffusion in fracture networks. Geophys. Res. Lett., 28(21): 4051-4054.] and Bodin et al. [Bodin, J., Porel, G. and Delay, F., 2003c. Simulation of solute transport in discrete fracture networks using the time domain random walk method. Earth Planet. Sci. Lett., 6566: 1-8.] for simulating solute transport in discrete fracture networks. It is assumed that the fracture network can reasonably be represented by a network of interconnected one-dimensional pipes (i.e. flow channels). Processes accounted for are: (1) advection and hydrodynamic dispersion in the channels, (2) matrix diffusion, (3) diffusion into stagnant zones within the fracture planes, (4) sorption reactions onto the fracture walls and in the matrix, (5) linear decay, and (6) mass sharing at fracture intersections. The TDRW method is handy and very efficient in terms of computation costs since it allows for the one-step calculation of the particle residence time in each bond of the network. This method has been programmed in C++, and efforts have been made to develop an efficient and user-friendly software, called SOLFRAC. This program is freely downloadable at the URL (labo.univ-poitiers.fr/hydrasa/intranet/telechargement.htm). It calculates solute transport into 2D pipe networks, while considering different types of injections and different concepts of local dispersion within each flow channel. Post-simulation analyses are also available, such as the mean velocity or the macroscopic dispersion at the scale of the entire network. The program may be used to evaluate how a given transport mechanism influences the macroscopic transport behaviour of fracture networks. It may also be used, as is the case, e.g., with analytical solutions, to interpret laboratory or field tracer test experiments performed in single fractures.     

Anomalous transport of colloids and solutes in a shear zone

Transport experiments with colloids and radionuclides in a shear zone were conducted during the Colloid and Radionuclide Retardation experiment (CRR) at Nagra's Grimsel Test Site. Breakthrough curves of bentonite colloids and uranine, a non-sorbing solute, were measured in an asymmetric dipole flow field. The colloid breakthrough is earlier than that of uranine. Both breakthrough curves show anomalously long late time tails and the slope of the late time tails for the colloids is slightly higher. Anomalous late time tails are commonly associated with matrix diffusion processes; the diffusive interaction of solutes transported in open channels with the adjacent porous rock matrix or zones of stagnant water. The breakthrough curves for different colloid size classes are very similar and show no signs of fractionation due to their (size-dependent) diffusivity. It is proposed that tailing of the colloids is mainly caused by the structure of the flow field and that for the colloid transport, matrix diffusion is of minor importance. This has consequences for the interpretation of the uranine breakthrough. Comparisons of experimental results with numerical studies and with the evaluation of the colloid breakthrough with continuous time random theory imply that the tailing in the conservative solute breakthrough in this shear zone is not only caused by matrix diffusion. Part of the tailing can be attributed to advective transport in fracture networks and advection in low velocity regions. Models based on the advection-dispersion equation and matrix diffusion do not properly describe the temporal and spatial evolution of colloid and solute transport in such systems with a consistent set of parameters.     

Modeling field-scale multiple tracer injection at a low-level waste disposal site in fractured rocks: effect of multiscale heterogeneity and source term uncertainty on conceptual understanding of mass transfer processes

Multiple factors may affect the scale-up of laboratory multi-tracer injection into structured porous media to the field. Under transient flow conditions and with multiscale heterogeneities in the field, previous attempts to scale-up laboratory experiments have not answered definitely the questions about the governing mechanisms and the spatial extent of the influence of small-scale mass transfer processes such as matrix diffusion. The objective of this research is to investigate the effects of multiscale heterogeneity, mechanistic and site model conceptualization, and source term density effect on elucidating and interpreting tracer movement in the field. Tracer release and monitoring information previously obtained in a field campaign of multiple, conservative tracer injection under natural hydraulic gradients at a low-level waste disposal site in eastern Tennessee, United States, is used for the research. A suite of two-pore-domain, or fracture-matrix, groundwater flow and transport models are calibrated and used to conduct model parameter and prediction uncertainty analyses. These efforts are facilitated by a novel nested Latin-hypercube sampling technique. Our results verify, at field scale, a multiple-pore-domain, multiscale mechanistic conceptual model that was used previously to interpret only laboratory observations. The results also suggest that, integrated over the entire field site, mass flux rates attributable to small-scale mass transfer are comparable to that of field-scale solute transport. The uncertainty analyses show that fracture spacing is the most important model parameter and model prediction uncertainty is relatively higher at the interface between the preferred flow path and its parent bedrock. The comparisons of site conceptual models indicate that the effect of matrix diffusion may be confined to the immediate neighborhood of the preferential flow path. Finally, because the relatively large amount of tracer needed for field studies, it is likely that source term density effect may exaggerate or obscure the effect of matrix diffusion on the movement of tracers from the preferred flow path into the bedrock.     

Numerical modeling of coupled variably saturated fluid flow and reactive transport with fast and slow chemical reactions

The couplings among chemical reaction rates, advective and diffusive transport in fractured media or soils, and changes in hydraulic properties due to precipitation and dissolution within fractures and in rock matrix are important for both nuclear waste disposal and remediation of contaminated sites. This paper describes the development and application of LEHGC2.0, a mechanistically based numerical model for simulation of coupled fluid flow and reactive chemical transport, including both fast and slow reactions in variably saturated media. Theoretical bases and numerical implementations are summarized, and two example problems are demonstrated. The first example deals with the effect of precipitation/dissolution on fluid flow and matrix diffusion in a two-dimensional fractured media. Because of the precipitation and decreased diffusion of solute from the fracture into the matrix, retardation in the fractured medium is not as large as the case wherein interactions between chemical reactions and transport are not considered. The second example focuses on a complicated but realistic advective-dispersive-reactive transport problem. This example exemplifies the need for innovative numerical algorithms to solve problems involving stiff geochemical reactions.     

Flow and transport in unsaturated fractured rock: effects of multiscale heterogeneity of hydrogeologic properties

The heterogeneity of hydrogeologic properties at different scales may have different effects on flow and transport processes in a subsurface system. A model for the unsaturated zone of Yucca Mountain, Nevada, is developed to represent complex heterogeneity at two different scales: (1) layer scale corresponding to geologic layering and (2) local scale. The layer-scale hydrogeologic properties are obtained using inverse modeling, based on the available measurements collected from the Yucca Mountain site. Calibration results show a significant lateral and vertical variability in matrix and fracture properties. Hydrogeologic property distributions in a two-dimensional, vertical cross-section of the site are generated by combining the average layer-scale matrix and fracture properties with local-scale perturbations generated using a stochastic simulation method. The unsaturated water flow and conservative (nonsorbing) tracer transport through the cross-section are simulated for different sets of matrix and fracture property fields. Comparison of simulation results indicates that the local-scale heterogeneity of matrix and fracture properties has a considerable effect on unsaturated flow processes, leading to fast flow paths in fractures and the matrix. These paths shorten the travel time of a conservative tracer from the source (repository) horizon in the unsaturated zone to the water table for small fractions of total released tracer mass. As a result, the local-scale heterogeneity also has a noticeable effect on global tracer transport processes, characterized by an average breakthrough curve at the water table, especially at the early arrival time of tracer mass. However, the effect is not significant at the later time after 20% tracer mass reaches the water table. The simulation results also verify that matrix diffusion plays an important role in overall solute transport processes in the unsaturated zone at Yucca Mountain.     

Field investigation into unsaturated flow and transport in a fault: model analyses

Results of a fault test performed in the unsaturated zone of Yucca Mountain, Nevada, were analyzed using a three-dimensional numerical model. The fault was explicitly represented as a discrete feature and the surrounding rock was treated as a dual-continuum (fracture-matrix) system. Model calibration against seepage and water-travel-velocity data suggests that lithophysal cavities connected to fractures can considerably enhance the effective fracture porosity and therefore retard water flow in fractures. Comparisons between simulation results and tracer concentration data also indicate that matrix diffusion is an important mechanism for solute transport in unsaturated fractured rock. We found that an increased fault-matrix and fracture-matrix interface areas were needed to match the observed tracer data, which is consistent with previous studies. The study results suggest that the current site-scale model for the unsaturated zone of Yucca Mountain may underestimate radionuclide transport time within the unsaturated zone, because an increased fracture-matrix interface area and the increased effective fracture porosity arising from lithophysal cavities are not considered in the current site-scale model.     

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.     

Analysis of the migration of instantaneously injected cesium in artificial fractures of Lac du Bonnet granite, Manitoba, Canada

Breakthrough curves of ¹³⁷Cs and tritiated water injected instantaneously into artificial fractures in Lac du Bonnet granite were analyzed using the analytical solution for a single rock-fracture system and assuming the linear sorption isotherm of the solute. Parameters of nuclide diffusion and sorption in rock matrices, obtained by fitting, varied depending on the flow velocity in the fractures. According to theoretical calculations, different fracture flow velocities lead to different diffusion distances of nuclides in matrices at the same injection volume. As microscopic inhomogeneity is considered to exist in the rock matrix, the average diffusion-sorption characteristics of the matrix within the diffusion distance may have varied with the fracture flow velocity. Surface sorption was marked in fractures that had relatively high matrix sorption-diffusion capacities. The phenomenon was interpreted using the theoretical relationships developed between the surface sorption, matrix sorption and pore diffusion coefficient, and the porosity of matrices.The effect of the nonlinear sorption of solute was examined by numerically solving model equations that incorporate the nonlinear isotherm. This incorporation may contribute to the reduction of deviations between theoretical and experimental BTC's.     

Applied tracer tests in fractured rock: Can we predict natural gradient solute transport more accurately than fracture and matrix parameters?

Applied tracer tests provide a means to estimate aquifer parameters in fractured rock. The traditional approach to analysing these tests has been using a single fracture model to find the parameter values that generate the best fit to the measured breakthrough curve. In many cases, the ultimate aim is to predict solute transport under the natural gradient. Usually, no confidence limits are placed on parameter values and the impact of parameter errors on predictions of solute transport is not discussed. The assumption inherent in this approach is that the parameters determined under forced conditions will enable prediction of solute transport under the natural gradient. This paper considers the parameter and prediction uncertainty that might arise from analysis of breakthrough curves obtained from forced gradient applied tracer tests. By adding noise to an exact solution for transport in a single fracture in a porous matrix we create multiple realisations of an initial breakthrough curve. A least squares fitting routine is used to obtain a fit to each realisation, yielding a range of parameter values rather than a single set of absolute values. The suite of parameters is then used to make predictions of solute transport under lower hydraulic gradients and the uncertainty of estimated parameters and subsequent predictions of solute transport is compared. The results of this study show that predictions of breakthrough curve characteristics (first inflection point time, peak arrival time and peak concentration) for groundwater flow speeds with orders of magnitude smaller than that at which a test is conducted can sometimes be determined even more accurately than the fracture and matrix parameters.     

Predictive modelling of dispersion controlled reactive plumes at the laboratory-scale

A model-based interpretation of laboratory-scale experimental data is presented. Hydrolysis experiments carried out using thin glass tanks filled with glass beads to construct a hypothetical and inert, homogeneous porous medium were analysed using a 2D numerical model. A new empirical formula, based upon results for non-reactive (tracer) experiments is used to calculate transversal dispersivity values for a range of grain sizes and any flow velocities. Combined with effective diffusion coefficients calculated from Stokes-Einstein type equations, plume lengths arising from mixing between two solutes can be predicted accurately using numerical modelling techniques. Moreover, pH and ion concentration profiles lateral to the direction of flow of the mixing species can be determined at any given point downstream, without the need for result fitting. In our case, this approach does not lead to overpredictions of lateral mixing, as previously reported when using parameters derived from non-reactive tracer experiments to describe reactive solute transport. The theory is based on the assumption of medium homogeneity.     

Matrix diffusion coefficients in volcanic rocks at the Nevada test site: influence of matrix porosity, matrix permeability, and fracture coating minerals

Diffusion cell experiments were conducted to measure nonsorbing solute matrix diffusion coefficients in forty-seven different volcanic rock matrix samples from eight different locations (with multiple depth intervals represented at several locations) at the Nevada Test Site. The solutes used in the experiments included bromide, iodide, pentafluorobenzoate (PFBA), and tritiated water ((3)HHO). The porosity and saturated permeability of most of the diffusion cell samples were measured to evaluate the correlation of these two variables with tracer matrix diffusion coefficients divided by the free-water diffusion coefficient (D(m)/D*). To investigate the influence of fracture coating minerals on matrix diffusion, ten of the diffusion cells represented paired samples from the same depth interval in which one sample contained a fracture surface with mineral coatings and the other sample consisted of only pure matrix. The log of (D(m)/D*) was found to be positively correlated with both the matrix porosity and the log of matrix permeability. A multiple linear regression analysis indicated that both parameters contributed significantly to the regression at the 95% confidence level. However, the log of the matrix diffusion coefficient was more highly-correlated with the log of matrix permeability than with matrix porosity, which suggests that matrix diffusion coefficients, like matrix permeabilities, have a greater dependence on the interconnectedness of matrix porosity than on the matrix porosity itself. The regression equation for the volcanic rocks was found to provide satisfactory predictions of log(D(m)/D*) for other types of rocks with similar ranges of matrix porosity and permeability as the volcanic rocks, but it did a poorer job predicting log(D(m)/D*) for rocks with lower porosities and/or permeabilities. The presence of mineral coatings on fracture walls did not appear to have a significant effect on matrix diffusion in the ten paired diffusion cell experiments.     

Radionuclide transport and retardation in rock fracture and crushed rock column experiments

Transport and retardation of non-sorbing tritiated water and chloride and slightly sorbing sodium was studied in Syyry area SY-KR7 mica gneiss, in altered porous tonalite and in fresh tonalite. Experiments were performed using dynamic fracture and crushed rock column methods. Static batch method for sodium was introduced to compare retardation values from static and dynamic experiments. The ¹⁴C-PMMA method was used to study the pore structure of matrices. The pore aperture distribution was evaluated from Hg-porosimetry determinations and the surface areas were determined using the B.E.T. method. The flow characteristics and transport behavior of tracers were interpreted using a numerical compartment model for dispersion. The effect of matrix diffusion was calculated using an analytical solution to the advection-matrix diffusion problem in which surface retardation was taken into account. Radionuclide transport behavior in rock fractures was explained on the basis of rock structure.     

Dual-domain solute transfer and transport processes: evaluation in batch and transport experiments

Soil macropore networks establish a dual-domain transport scenario in which water and solutes are preferentially channeled through soil macropores while slowly diffusing into and out of the bulk soil matrix. The influence of macropore networks on intra-ped solute diffusion and preferential transport in a soil typical of subsurface-drained croplands in the Midwestern United States was studied in batch- and column-scale experiments. In the batch diffusion studies with soil aggregates, the estimated diffusion radius (length) of the soil aggregates corresponded to the half-spacing of the aggregate fissures, suggesting that the intra-ped fissures reduced the diffusion impedance and preferentially allowed solutes to diffuse into the soil matrix. In the column-scale solute transport experiments, the average diffusion radius (estimated from HYDRUS-2D simulations and a first-order diffusive transfer term) was nearly double that of the batch-scale study. This increase may be attributed to a loss of pore continuity and a compounding of the small diffusion impedance through macropores at the larger scale. The column-scale solute transport experiments also suggest that two preferential networks exist in the soil. At and near soil saturation, a primary network of large macropores (possibly root channels and earthworm burrows) dominate advective transport, causing a high degree of physical and sorption nonequilibrium and simultaneous breakthrough of a nonreactive (bromide) and a reactive (alachlor) solute. As the saturation level decreases, the primary network drains, while transport through smaller macropores (possibly intra-ped features) continues, resulting in a reduced degree of nonequilibrium and separation in the breakthrough curves of bromide and alachlor.     

Laboratory analog and numerical study of groundwater flow and solute transport in a karst aquifer with conduit and matrix domains

New mathematical and laboratory methods have been developed for simulating groundwater flow and solute transport in karst aquifers having conduits imbedded in a porous medium, such as limestone. The Stokes equations are used to model the flow in the conduits and the Darcy equation is used for the flow in the matrix. The Beavers–Joseph interface boundary conditions are adopted to describe the flow exchange at the interface boundary between the two domains. A laboratory analog is used to simulate the conduit and matrix domains of a karst aquifer. The conduit domain is located at the bottom of the transparent plexiglas laboratory analog and glass beads occupy the remaining space to represent the matrix domain. Water flows into and out of the two domains separately and each has its own supply and outflow reservoirs. Water and solute are exchanged through an interface between the two domains. Pressure transducers located within the matrix and conduit domains of the analog provide data that is processed and stored in digital format. Dye tracing experiments are recorded using time-lapse imaging. The data and images produced are analyzed by a spatial analysis program. The experiments provide not only hydraulic head distribution but also capture solute front images and mass exchange measurements between the conduit and matrix domains. In the experiment, we measure and record pressures, and quantify flow rates and solute transport. The results present a plausible argument that laboratory analogs can characterize groundwater water flow, solute transport, and mass exchange between the conduit and matrix domains in a karst aquifer. The analog validates the predictions of a numerical model and demonstrates the need of laboratory analogs to provide verification of proposed theories and the calibration of mathematical models.     

Modelling the solute transport under nonequilibrium conditions on the basis of mass transfer equations

A solute transport model that describes nonequilibrium adsorption in soil/groundwater systems by mass transfer equations for film and intraparticle diffusion is presented. The model is useful in cases where breakthrough curve spreading cannot be explained by dispersion only. To evaluate its validity, the model was applied to several data sets from column experiments. The validity was also proved by a comparison with an analytical solution for the limiting case of predominating dispersion. Furthermore, a sensitivity analysis was performed to illustrate the influence of different process and sorption parameters (pore water velocity, intraparticle mass transfer coefficient, isotherm nonlinearity) on the shape of the calculated breakthrough curves. The application of the proposed model is discussed in comparison to the widely used dispersed flow/local equilibrium model, and a relationship between both models, which is based on a lumped parameter approach, is shown.