Harihar Rajaram

Solute Transport in Variable Aperture Fractures: An Investigation of the Relative Importance of Taylor Dispersion and Macrodispersion

Dispersion of solutes in a variable aperture fracture results from a combination of molecular diffusion and velocity variations in both the plane of the fracture (macrodispersion) and across the fracture aperture (Taylor dispersion). We use a combination of physical experiments and computational simulations to test a theoretical model in which the effective longitudinal dispersion coefficient DL is expressed as a sum of the contributions of these three dispersive mechanisms. The combined influence of Taylor dispersion and macrodispersion results in a nonlinear dependence of DL on the Peclet number (Pe 5 V^b&/Dm, where V is the mean solute velocity, ^b& is the mean aperture, and Dm is the molecular diffusion coefficient). Three distinct dispersion regimes become evident: For small Pe (Pe , , 1), molecular diffusion dominates resulting in D L } Pe 0 ; for intermediate Pe, macrodispersion dominates (DL } Pe); and for large Pe, Taylor dispersion dominates (D L } Pe 2 ). The Pe range corresponding to these different regimes is controlled by the statistics of the aperture field. In particular, the upper limit of Pe corresponding to the macrodispersion regime increases as the macrodispersivity increases. Physical experiments in an analog, rough-walled fracture confirm the nonlinear Pe dependence of DL predicted by the theoretical model. However, the theoretical model underestimates the magnitude of DL. Computational simulations, using a particle-tracking algorithm that incorporates all three dispersive mechanisms, agree very closely with the theoretical model predictions. The close agreement between the theoretical model and computational simulations is largely because, in both cases, the Reynolds equation describes the flow field in the fracture. The discrepancy between theoretical model predictions and DL estimated from the physical experiments appears to be largely due to deviations from the local cubic law assumed by the Reynolds equation.

Saturated flow in a single fracture: evaluation of the Reynolds Equation in measured aperture fields

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

Influence of aperture variability on dissolutional growth of fissures in Karst Formations

The influence of aperture variability on dissolutional growth of fissures is investigated on the basis of two-dimensional numerical simulations. The logarithm of the fissure aperture before dissolution begins is modeled as a Gaussian stationary isotropic random field. The initial phase of dissolutional growth is studied up to the time when turbulent flow first occurs at a point within the fissure (the breakthrough time). The breakthrough time in variable aperture fissures is smaller than that in uniform fissures and decreases as the coefficient of variation of the aperture field (σ/μ) increases. In comparing uniform and variable aperture fissures in limestone, the breakthrough time with σ/μ=0.1 is about a factor of 2 smaller than that in a uniform fissure. The breakthrough time is reduced by about an order of magnitude with σ/μ=2.0. The mechanism leading to reduced breakthrough times is the focusing of flow into preferential flow channels which are enlarged at a faster rate than the surrounding regions of slower flow. Dissolution channels are narrower and more tortuous as σ/μ increases. Investigations of the influence of reaction rate reveal that the influence of aperture variability is more pronounced in rapidly dissolving rock. In uniform fissures in rapidly dissolving minerals, breakthrough times are very long since water becomes saturated with respect to the mineral within a short distance of the entrance to the flow path. However, in variable aperture fissures, breakthrough occurs rapidly because of selective growth along preferential flow channels, which progressively capture larger fractions of the total flow. These results partly explain why conduits develop rapidly in gypsum, although previous one-dimensional studies suggest that conduit growth will not occur.

Plume scale‐dependent dispersion in heterogeneous aquifers: 2. Eulerian analysis and three‐dimensional aquifers

An analytical approach is developed for describing the ensemble average of the second moment of a solute plume in three-dimensional heterogeneous porous media. While existing approaches describe scale-dependent dispersion in terms of a single scale, the plume displacement, the approach developed here presents an enhanced picture of scale-dependent dispersion involving two scales: the plume displacement and the plume scale. The plume scale arises naturally in the formulation, permitting a distinction between the dispersive role of heterogeneity at scales smaller than the plume size and the variability in the plume location caused by larger scale heterogeneity. A physically consistent description of scale-dependent dispersion is thus achieved. The growth of the ensemble average second moment is related to the product of concentration values at two points. The concept of the separation distribution function related to the latter is introduced. The separation distribution function physically describes the fraction of solute particles which have another solute particle at a given separation. An Eulerian partial differential equation based on a small perturbation approach is developed to describe the evolution of the separation distribution function. Simple analytical expressions for the second moment growth rates are presented. These expressions incorporate the influence of the plume size through a low wavenumber filter depending of the plume second moment. Asymptotic expressions for the second moment growth rate are presented which apply at large displacement. These expressions indicate that the longitudinal second moment growth rate depends on the transverse second moments of the plume. Comparison of predicted second moment evolution with results from earlier numerical simulations indicates excellent agreement. Application to the Borden tracer test indicates a significant reduction in the longitudinal second moment from that predicted by existing three-dimensional theories and better agreement with the measured second moments.

Plume-Scale Dependent Dispersion in Aquifers with a Wide Range of Scales of Heterogeneity

Dispersivity computations are developed for heterogeneous aquifers which exhibit structured variability across a wide range of scales. Multiscale, fractional Gaussian and self-similar random field models are used to describe the statistical structure of hydraulic conductivity variations. The concept of a relative dispersivity (Aijr) is introduced, following a similar concept introduced earlier in the context of turbulent diffusion. The relative dispersivity depends on the plume size and is influenced only by velocity variations at scales smaller than the plume scale. In contrast, the dispersivity associated with the ensemble average concentration (Aij) is influenced by all scales of velocity variations, including very large scales. In fractional Gaussian media, previous analyses predict that Aij grows as a power function of the plume displacement. In contrast, the longitudinal relative dispersivity approaches a constant value, and the transverse components gradually approach zero. In fractional Gaussian and self-similar media the longitudinal dispersivity is shown to be a power function of the transverse second moment of the plume. The initial plume dimensions have a strong influence on dispersivity values for plumes in media with structured variability across a wide range of scales. A generalization of self-similar random field models is presented to accommodate the widely observed feature of statistical anisotropy in ln K variability. A practical illustration of the results is presented based on ln K variability data at the Cape Cod site. Two-scale exponential and self-similar models are fit to the ln K variograms, and dispersivity computations are developed for two plumes with very different initial dimensions. At a displacement of about 1 km, a longitudinal relative dispersivity lpar;A11r) value of 30 m is estimated for the sewage plume with initial dimensions of 500 × 500 × 10 m, while the estimate corresponding to the tracer test plume with initial dimensions of 5 × 5 × 1 m is about 1–2 m. The corresponding ensemble longitudinal dispersivity (A11) is about 90 m. Relative dispersivities are more appropriate for characterizing the dilution and spreading at individual heterogeneous aquifers. Dispersivities associated with the ensemble average concentration will tend to overestimate the degree of dilution and spreading in an aquifer, and this error can be very large in media with a wide range of scales of heterogeneity.

Three‐dimensional spatial moments analysis of the Borden Tracer Test

The large-scale tracer test conducted at the Borden site included detailed three-dimensional plume monitoring, but most previous studies have emphasized the interpretation of the data in a two-dimensional sense. Here, an independent analysis of the fully three-dimensional nature of the plume is developed using a linear spatial interpolation scheme which differs significantly from the gridded higher-order interpolation scheme used in earlier analyses. A detailed examination of the data base reveals significant truncation of the plume during several sampling sessions. This feature explains the low mass estimates obtained during the later sampling sessions. Consequently, there are significant uncertainties associated with the corresponding second-moment estimates. This uncertainty, which calls into question any refined comparisons of second-moment evolution with stochastic theories, has not been recognized in previous interpretations of the second-moment estimates. The estimated vertical macrodispersivity is found to be about twice that attributable to molecular diffusion. The estimated horizontal transverse macrodispersivity is much larger than the vertical macrodispersivity. Temporal variations in the flow appear to provide a plausible explanation of the large horizontal transverse macrodispersivity. Anamolous behavior of the second-moment evolution suggests the role of a large-scale heterogeneity during the later portion of the test, indicating possible mutual interactions between the second-moment components.

Modeling hyporheic zone processes

Stream biogeochemistry is influenced by the physical and chemical processes that occur in the surrounding watershed. These processes include the mass loading of solutes from terrestrial and atmospheric sources, the physical transport of solutes within the watershed, and the transformation of solutes due to biogeochemical reactions. Research over the last two decades has identified the hyporheic zone as an important part of the stream system in which these processes occur. The hyporheic zone may be loosely defined as the porous areas of the stream bed and stream bank in which stream water mixes with shallow groundwater. Exchange of water and solutes between the stream proper and the hyporheic zone has many biogeochemical implications, due to differences in the chemical composition of surface and groundwater. For example, surface waters are typically oxidized environments with relatively high dissolved oxygen concentrations. In contrast, reducing conditions are often present in groundwater systems leading to low dissolved oxygen concentrations. Further, microbial oxidation of organic materials in groundwater leads to supersaturated concentrations of dissolved carbon dioxide relative to the atmosphere. Differences in surface and groundwater pH and temperature are also common. The hyporheic zone is therefore a mixing zone in which there are gradients in the concentrations of dissolved gasses, the concentrations of oxidized and reduced species, pH, and temperature. These gradients lead to biogeochemical reactions that ultimately affect stream water quality. Due to the complexity of these natural systems, modeling techniques are frequently employed to quantify process dynamics. This special issue of Advances in Water Resources presents recent research on the modeling of hyporheic zone processes. To begin this preface, a brief history on modeling hyporheic zone processes is presented. This background information is by no means complete; additional information may be found in Streams and Groundwaters [17] and the references therein. The preface concludes with an overview of current re-

Plume scale‐dependent dispersion in heterogeneous aquifers: 1. Lagrangian analysis in a stratified aquifer

An analytical approach is developed for describing the ensemble average of the second moment of a solute plume in heterogeneous porous media. The growth of the ensemble average second moment is related to the growth of the mean square separation of a pair of particles. Exact Lagrangian expressions are developed for the growth of the mean square separation in a perfectly stratified aquifer. These exact expressions are made possible by an exact relation between the Lagrangian and Eulerian velocity covariances in the perfectly stratified aquifer. The ensemble average second moment is shown to depend on the initial vertical dimensions of the concentration distribution. As the vertical size increases, the second moment growth rate is larger. The second moment expressions are contrasted with the expressions for the mean square displacement of a single particle. While all scales of heterogeneity contribute to the mean square displacement of a single particle, only scales of heterogeneity smaller than the plume size contribute to the second moment. Asymptotic large displacement expressions for the case of a Gaussian covariance function describing the hydraulic conductivity variations are derived. These expressions indicate that at large displacement, the second moment grows as the 3/2 power of displacement. However, for any finite source size no matter how large, the second moment growth rate approaches a different asymptote from the rate of growth of the mean square displacement of a single particle. This is in conflict with the traditional notion of ergodicity which leads to the expectation that for large source sizes, the dispersivity approaches the rate of growth of the mean square displacement of a single particle.

Predictive modeling of flow and transport in a two‐dimensional intermediate‐scale, heterogeneous porous medium

As a first step toward understanding the role of sedimentary structures in flow and transport through porous media, this work deterministically examines how small-scale laboratory-measured values of hydraulic conductivity relate to in situ values of simple, artificial structures in an intermediate-scale (10 m long), two-dimensional, heterogeneous, laboratory experiment. Results were judged based on how well simulations using measured values of hydraulic conductivities matched measured hydraulic heads, net flow, and transport through the tank. Discrepancies were investigated using sensitivity analysis and nonlinear regression estimates of the in situ hydraulic conductivity that produce the best fit to measured hydraulic heads and net flow. Permeameter and column experiments produced laboratory measurements of hydraulic conductivity for each of the sands used in the intermediate-scale experiments. Despite explicit numerical representation of the heterogeneity the laboratory-measured values underestimated net flow by 12–14% and were distinctly smaller than the regression-estimated values. The significance of differences in measured hydraulic conductivity values was investigated by comparing variability of transport predictions using the different measurement methods to that produced by different realizations of the heterogeneous distribution. Results indicate that the variations in measured hydraulic conductivity were more important to transport than variations between realizations of the heterogeneous distribution of hydraulic conductivity.

Intermediate-scale experiments and numerical simulations of transport under radial flow in a two-dimensional heterogeneous porous medium.

Estimated field dispersivities from uniform and nonuniform flow tracer tests are often compared without distinction, yet the difference in the flow field is important. The difference is investigated here based on intermediate-scale experiments and numerical simulations in two-dimensional heterogeneous porous media. The scale dependence of dispersivities estimated from radial flow tracer tests in two-dimensional heterogeneous porous media and the variability of these estimates are quantified. The results presented in this paper demonstrate the difference between the scale dependence inferred from uniform flow and radial flow tracer experiments in the same random hydraulic conductivity realization. In particular, dispersivities estimated using type-curve matching from radial flow tracer experiments continue to exhibit a scale dependence, even at scales where an asymptotic constant dispersivity value applies for transport in a uniform mean flow. The discrepancy between the behavior of transport in uniform and radial flows is partly due to the converging nature of the radial flow and, more importantly, due to the small source size involved in forced-gradient tracer tests. There is substantial variability in dispersivity values estimated from different injection points at the same radial distance from the pumping well. In the range of σlnK = 1.0 to 2.5 the coefficient of variation of the dispersivity estimated at the same radial distance approaches a value of about 1.0 at radial distances much larger than the correlation scale.