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Field determination of the three-dimensional hydraulic conductivity tensor of anisotropic media: 1. Theory

A field method is proposed for determining the three-dimensional hydraulic conductivity tensor and specific storage of an anisotropic porous or fractured medium. The method, known as cross-hole testing (to distinguish it from conventional single-hole packer tests), consists of injecting fluid into (or withdrawing fluid from) packed-off intervals in a number of boreholes and monitoring the transient head response in similar intervals in neighboring boreholes. The directions of the principal hydraulic conductivities need not be known prior to the test, and the boreholes may have arbitrary orientations (e.g., they can all be vertical). An important aspect of the proposed method is that it provides direct field information on whether it is proper to regard the medium as being uniform and anisotropic on the scale of the test. The first paper presents theoretical expressions describing transient and steady state head response in monitoring intervals of arbitrary lengths and orientations, to constant-rate injection into (or withdrawal from) intervals having similar or different lengths and orientations. The conditions under which these intervals can be treated mathematically as points are investigated by an asymptotic analysis. The effect of planar no-flow and constant-head boundaries on the response is analyzed by the theory of images. The second paper describes the field methodology and shows how the proposed approach works in the case of fractured granitic rocks.

Water Resources Research

Field determination of the three-dimensional hydraulic conductivity tensor of anisotropic media: 2. Methodology and application to fractured rocks

The analytical solutions developed in the first paper can be used to interpret the results of cross-hole tests conducted in anisotropic porous or fractured media. In the particular case where the injection and monitoring intervals are short relative to the distance between them, the test results can be analyzed graphically. From the transient variation of hydraulic head in a given monitoring interval, one can determine the directional hydraulic diffusivity, K d ( e )/ S s , and the quantity D / S s , by curve matching. (Here K d ( e ) is directional hydraulic conductivity parallel to the unit vector, e , pointing from the injection to the monitoring interval, S s is specific storage, and D is the determinant of the hydraulic conductivity tensor, K .) The principal values and directions of K , together with S s , can then be evaluated by fitting an ellipsoid to the square roots of the directional diffusivities. Ideally, six directional measurements are required. In practice, a larger number of measurements is often necessary to enable fitting an ellipsoid to the data by least squares. If the computed [ K d ( e )/ s s ] ½ values fluctuate so severely that a meaningful least squares fit is not possible, one has a direct indication that the subsurface does not behave as a uniform anisotropic medium on the scale of the test. Test results from a granitic rock near Oracle in southern Arizona are presented to illustrate how the method works for fractured rocks. At the site, the Oracle granite is shown to respond as a near-uniform, anisotropic medium, the hydraulic conductivity of which is strongly controlled by the orientations of major fracture sets. The cross-hole test results are shown to be consistent with the results of more than 100 single-hole packer tests conducted at the site.

Water Resources Research

Assessment of the instantaneous unit hydrograph derived from the theory of topologically random networks

An instantaneous unit hydrograph (iuh) based on the theory of topologically random networks (topological iuh) is evaluated in terms of sets of basin characteristics and hydraulic parameters. Hydrographs were computed using two linear routing methods for each of two drainage basins in the southeastern United States and are the basis of comparison for the topological iuh's. Elements in the sets of basin characteristics for the topological iuh's are the number of first-order streams only, ( N ), or the nuber of sources together with the number of channel links in the topological diameter ( N, D ); the hydraulic parameters are values of the celerity and diffusivity constant. Sensitivity analyses indicate that the mean celerity of the internal links in the network is the critical hydraulic parameter for determining the shape of the topological iuh, while the diffusivity constant has minimal effect on the topological iuh. Asymptotic results (source-only) indicate the number of sources need not be large to approximate the topological iuh with the Weibull probability density function.

Water Resources Research

A policy evaluation tool: Management of a multiaquifer system using controlled stream recharge

A model for the optimal allocation of water resources was developed for a multiaquifer groundwater and surface water system near Livermore, California. The complex groundwater system was analyzed using a transient, quasi-three-dimensional model that considers the nonlinear behavior of the unconfined aquifer. The surface water system consists of a reservoir that discharges water to three streams which in turn recharge the upper aquifer. Nonlinear streamflow-recharge relationships were developed based upon synoptic field measurements of streamflow. The management model uses constrained optimization to minimize the cost of allocating surface water subject to physical and economic restrictions. Results indicate that a combined hydrologic and economic management model can be used to evaluate management practices of a complex hydrogeologic system. Questions can be posed which either would be impossible or extremely difficult to solve without the management model. We demonstrate the utility of such a model in three areas. First, the efficiency of intra-basin water allocations is evaluated. Second, critical factors that control management decisions of the basin are identified. Third, the influence of economic incentives that can best satisfy the conflicting objectives of various water users is explored.

Water Resources Research

Estimation of distributional parameters for censored trace level water quality data: 1. Estimation techniques

A recurring difficulty encountered in investigations of many metals and organic contaminants in ambient waters is that a substantial portion of water sample concentrations are below limits of detection established by analytical laboratories. Several methods were evaluated for estimating distributional parameters for such censored data sets using only uncensored observations. Their reliabilities were evaluated by a Monte Carlo experiment in which small samples were generated from a wide range of parent distributions and censored at varying levels. Eight methods were used to estimate the mean, standard deviation, median, and interquartile range. Criteria were developed, based on the distribution of uncensored observations, for determining the best performing parameter estimation method for any particular data set. The most robust method for minimizing error in censored-sample estimates of the four distributional parameters over all simulation conditions was the log-probability regression method. With this method, censored observations are assumed to follow the zero-to-censoring level portion of a lognormal distribution obtained by a least squares regression between logarithms of uncensored concentration observations and their z scores. When method performance was separately evaluated for each distributional parameter over all simulation conditions, the log-probability regression method still had the smallest errors for the mean and standard deviation, but the lognormal maximum likelihood method had the smallest errors for the median and interquartile range. When data sets were classified prior to parameter estimation into groups reflecting their probable parent distributions, the ranking of estimation methods was similar, but the accuracy of error estimates was markedly improved over those without classification.

Water Resources Research

Estimation of distributional parameters for censored trace level water quality data: 2. Verification and applications

Estimates of distributional parameters (mean, standard deviation, median, interquartile range) are often desired for data sets containing censored observations. Eight methods for estimating these parameters have been evaluated by R. J. Gilliom and D. R. Helsel (this issue) using Monte Carlo simulations. To verify those findings, the same methods are now applied to actual water quality data. The best method (lowest root-mean-squared error (rmse)) over all parameters, sample sizes, and censoring levels is log probability regression (LR), the method found best in the Monte Carlo simulations. Best methods for estimating moment or percentile parameters separately are also identical to the simulations. Reliability of these estimates can be expressed as confidence intervals using rmse and bias values taken from the simulation results. Finally, a new simulation study shows that best methods for estimating uncensored sample statistics from censored data sets are identical to those for estimating population parameters. Thus this study and the companion study by Gilliom and Helsel form the basis for making the best possible estimates of either population parameters or sample statistics from censored water quality data, and for assessments of their reliability.

Water Resources Research

Modeling contamination of shallow unconfined aquifers through infiltration beds

We model the transport of a simply reactive contaminant through an infiltration bed and underlying shallow, one-dimensional, unconfined aquifer with a plane, steeply sloping bottom in the assumed absence of dispersion and downgradient dilution. The effluent discharge and ambient groundwater flow under the infiltration beds are presumed to form a vertically mixed plume marked by an appreciable radial velocity component in the near field flow region. The near field analysis routes effluent contamination as a single linear reservoir whose output forms a source plane for the one-dimensional, far field flow region downgradient of the facility; the location and width of the source plane reflect the relative strengths of ambient flow and effluent discharge. We model far field contaminant transport, using an existing method of characteristics solution with frame speeds modified by recharge, bottom slope, and linear adsorption, and concentrations reflecting first-order reaction kinetics. The near and far field models simulate transport of synthetic detergents, chloride, total nitrogen, and boron in a contaminant plume at the Otis Air Force Base sewage treatment plant in Barnstable County, Massachusetts, with reasonable accuracy.

Massachusetts

Groundwater transport of strontium 90 in a glacial outwash environment

As part of the investigation of groundwater contamination at a uranium-scrap recovery plant at Wood River Junction, Rhode Island, laboratory experiments led to the development of a model for predicting the transport of strontium 90 in glacial outwash sediments based on an approximate mechanism for ion exchange. The multicomponent system was simplified to two components by regarding all exchangeable cations other than strontium 90 as a single component. The binary ion-exchange parameter was a function of the variable, total ion concentration. A one-dimensional solute transport model was formulated to evaluate the time necessary for natural groundwater flow to remove the strontium 90 contamination plume from the groundwater system to the Pawcatuck River. The finite difference transport equations were solved sequentially for total ion concentrations, then strontium 90 concentrations. Clay-free quartz and feldspar sands at the study site have little potential for strontium 90 sorption, and high calcium, magnesium, and sodium concentrations compete for the few ion exchange sites. As the total ion concentration plume moves out of the system, ion exchange of strontium 90 increases, reducing the strontium 90 concentration in the groundwater. Cleanout times predicted using the binary ion exchange mechanism were about two thirds of those predicted using a constant distribution coefficient. It is suggested that this type of model can simulate solute transport more realistically in many groundwater systems where the total ion concentration is not constant.

Rhode Island

Stochastic analysis of three-dimensional flow in a bounded domain

A commonly accepted first-order approximation of the equation for steady state flow in a fully saturated spatially random medium has the form of Poisson's equation. This form allows for the advantageous use of Green's functions to solve for the random output (hydraulic heads) in terms of a convolution over the random input (the logarithm of hydraulic conductivity). A solution for steady state three- dimensional flow in an aquifer bounded above and below is presented; consideration of these boundaries is made possible by use of Green's functions to solve Poisson's equation. Within the bounded domain the medium hydraulic conductivity is assumed to be a second-order stationary random process as represented by a simple three-dimensional covariance function. Upper and lower boundaries are taken to be no-flow boundaries; the mean flow vector lies entirely in the horizontal dimensions. The resulting hydraulic head covariance function exhibits nonstationary effects resulting from the imposition of boundary conditions. Comparisons are made with existing infinite domain solutions.

Water Resources Research

The problem of complex eigensystems in the semianalytical solution for advancement of time in solute transport simulations: a new method using real arithmetic

In the numerical modeling of groundwater solute transport, explicit solutions may be obtained for the concentration field at any future time without computing concentrations at intermediate times. The spatial variables are discretized and time is left continuous in the governing differential equation. These semianalytical solutions have been presented in the literature and involve the eigensystem of a coefficient matrix. This eigensystem may be complex (i.e., have imaginary components) due to the asymmetry created by the advection term in the governing advection-dispersion equation. Previous investigators have either used complex arithmetic to represent a complex eigensystem or chosen large dispersivity values for which the imaginary components of the complex eigenvalues may be ignored without significant error. It is shown here that the error due to ignoring the imaginary components of complex eigenvalues is large for small dispersivity values. A new algorithm that represents the complex eigensystem by converting it to a real eigensystem is presented. The method requires only real arithmetic.

Water Resources Research

Ground-water flow in low permeability environments

Certain geologic media are known to have small permeability; subsurface environments composed of these media and lacking well developed secondary permeability have groundwater flow sytems with many distinctive characteristics. Moreover, groundwater flow in these environments appears to influence the evolution of certain hydrologic, geologic, and geochemical systems, may affect the accumulation of pertroleum and ores, and probably has a role in the structural evolution of parts of the crust. Such environments are also important in the context of waste disposal. This review attempts to synthesize the diverse contributions of various disciplines to the problem of flow in low-permeability environments. Problems hindering analysis are enumerated together with suggested approaches to overcoming them. A common thread running through the discussion is the significance of size- and time-scale limitations of the ability to directly observe flow behavior and make measurements of parameters. These limitations have resulted in rather distinct small- and large-scale approaches to the problem. The first part of the review considers experimental investigations of low-permeability flow, including in situ testing; these are generally conducted on temporal and spatial scales which are relatively small compared with those of interest. Results from this work have provided increasingly detailed information about many aspects of the flow but leave certain questions unanswered. Recent advances in laboratory and in situ testing techniques have permitted measurements of permeability and storage properties in progressively “tighter” media and investigation of transient flow under these conditions. However, very large hydraulic gradients are still required for the tests; an observational gap exists for typical in situ gradients. The applicability of Darcy's law in this range is therefore untested, although claims of observed non-Darcian behavior appear flawed. Two important nonhydraulic flow phenomena, osmosis and ultrafiltration, are experimentally well established in prepared clays but have been incompletely investigated, particularly in undisturbed geologic media. Small-scale experimental results form much of the basis for analyses of flow in low-permeability environments which occurs on scales of time and size too large to permit direct observation. Such large-scale flow behavior is the focus of the second part of the review. Extrapolation of small-scale experimental experience becomes an important and sometimes controversial problem in this context. In large flow systems under steady state conditions the regional permeability can sometimes be determined, but systems with transient flow are more difficult to analyze. The complexity of the problem is enhanced by the sensitivity of large-scale flow to the effects of slow geologic processes. One-dimensional studies have begun to elucidate how simple burial or exhumation can generate transient flow conditions by changing the state of stress and temperature and by burial metamorphism. Investigation of the more complex problem of the interaction of geologic processes and flow in two and three dimensions is just beginning. Because these transient flow analyses have largely been based on flow in experimental scale systems or in relatively permeable systems, deformation in response to effective stress changes is generally treated as linearly elastic; however, this treatment creates difficulties for the long periods of interest because viscoelastic deformation is probably significant. Also, large-scale flow simulations in argillaceous environments generally have neglected osmosis and ultrafiltration, in part because extrapolation of laboratory experience with coupled flow to large scales under in situ conditions is controversial. Nevertheless, the effects are potentially quite important because the coupled flow might cause ultra long lived transient conditions. The difficulties associated with analysis are matched by those of characterizing hydrologic conditions in tight environments; measurements of hydraulic head and sampling of pore fluids have been done only rarely because of the practical difficulties involved. These problems are also discussed in the second part of this paper.

Water Resources Research

Groundwater Seepage Vectors and the Potential for Hillslope Failure and Debris Flow Mobilization

Insight for understanding the effect of groundwater flow on the potential for hillslope failure and liquefaction is provided by a novel limit‐equilibrium analysis of infinite slopes with steady, uniform Darcian seepage of arbitrary magnitude and direction. Normalization of the limit‐equilibrium solution shows that three dimensionless parameters govern completely the Coulomb failure potential of saturated, cohesionless, infinite homogeneous hillslopes: (1) the ratio of seepage force magnitude to gravitational body force magnitude; (2) the angle θ − Φ, where θ is the surface slope angle and Φ is the angle of internal friction of the soil; and (3) the angle λ + Φ, where λ is the angle of the seepage vector measured with respect to an outward‐directed surface‐normal vector. An additional dimensionless parameter affects the solution if soil cohesion is included in the analysis. Representation of the normalized solution as a single family of curves shows that minimum slope stability universally occurs when the seepage direction is given by λ = 90° − Φ. It also shows that for some upward seepage conditions, slope stability is limited by static liquefaction rather than by Coulomb failure. Close association between these liquefaction conditions and certain Coulomb failure conditions indicates that slope failure in such instances could be responsible for nearly spontaneous mobilization of destructive flowing soil masses on hillslopes.

Water Resources Research

A new formula for the analytical solution of the radial dispersion problem

A new formula is presented for the analytical solution of the radial dispersion problem, which analyzes the dispersive transport of a tracer in radial flow from a recharge well. This formula is simpler than a solution presented by previous investigators, but the two solutions are shown to be equivalent. Because the analytical solution consists of an integral of an oscillatory function, it is evaluated as an alternating series, the terms of which are the integrals over successive half cycles of the integrand. An efficient algorithm is used to accelerate convergence of the series. The accuracies of previous solutions obtained by finite difference, numerical Laplace transform, and perturbation methods are examined by comparison with the analytical solution. Numerical values of the solution are tabulated for readers interested in checking the accuracies of solute transport codes.

Water Resources Research

On tide-induced Lagrangian residual current and residual transport: 1. Lagrangian residual current

Residual currents in tidal estuaries and coastal embayments have been recognized as fundamental factors which affect the long-term transport processes. It has been pointed out by previous studies that it is more relevant to use a Lagrangian mean velocity than an Eulerian mean velocity to determine the movements of water masses. Under weakly nonlinear approximation, the parameter k , which is the ratio of the net displacement of a labeled water mass in one tidal cycle to the tidal excursion, is assumed to be small. Solutions for tides, tidal current, and residual current have been considered for two-dimensional, barotropic estuaries and coastal seas. Particular attention has been paid to the distinction between the Lagrangian and Eulerian residual currents. When k is small, the first-order Lagrangian residual is shown to be the sum of the Eulerian residual current and the Stokes drift. The Lagrangian residual drift velocity or the second-order Lagrangian residual current has been shown to be dependent on the phase of tidal current. The Lagrangian drift velocity is induced by nonlinear interactions between tides, tidal currents, and the first-order residual currents, and it takes the form of an ellipse on a hodograph plane. Several examples are given to further demonstrate the unique properties of the Lagrangian residual current.

Water Resources Research

Unsaturated flow in a centrifugal field: Measurement of hydraulic conductivity and testing of Darcy's Law

A method has been developed to establish steady state flow of water in an unsaturated soil sample spinning in a centrifuge. Theoretical analysis predicts moisture conditions in the sample that depend strongly on soil type and certain operating parameters. For Oakley sand, measurements of flux, water content, and matric potential during and after centrifugation verify that steady state flow can be achieved. Experiments have confirmed the theoretical prediction of a nearly uniform moisture distribution for this medium and have demonstrated that the flow can be effectively one-dimensional. The method was used for steady state measurements of hydraulic conductivity K for relatively dry soil, giving values as low as 7.6 × 10 −11 m/s with data obtained in a few hours. Darcy's law was tested by measuring K for different centrifugal driving forces but with the same water content. For the sand at a bulk density of 1.82 Mg/m 3 and 27% saturation, results were consistent with Darcy's law for K equal to 5.22 × 10 −10 m/s and forces ranging from 216 to 1650 times normal gravity.

Water Resources Research

A compositional multiphase model for groundwater contamination by petroleum products: 1. Theoretical considerations

A mathematical model is developed to describe the fate of hydrocarbon constituents of petroleum products introduced to soils as an immiscible liquid from sources such as leaking underground storage tanks and ruptured pipelines. The problem is one of multiphase transport (oil (immiscible), air, and water phases) of a reactive contaminant with constituents such as benzene, toluene, and xylene found in refined petroleum products like gasoline. In the unsaturated zone, transport of each constituent can occur as a solute in the water phase, vapor in the air phase, and as an unaltered constituent in the oil phase. Additionally, the model allows for adsorption. Molecular transformations, microbially mediated or abiotic, are incorporated as sink terms in the conservation of mass equations. An equilibrium approximation, applicable to any immiscible organic contaminant is applied to partition constituent mass between the air, oil, water, and adsorbed phases for points in the region where the oil phase exists. Outside the oil plume the equilibrium approximation takes on a simpler form to partition constituent mass between the air, water, and adsorbed phases only. Microbial degradation of petroleum products is first discussed in a general model, then the conservation of mass equation for oxygen is incorporated into the analysis which takes advantage of the key role played by oxygen in the metabolism of hydrocarbon utilizing microbes in soil environments. Approximations to two subproblems, oil plume establishment in the unsaturated zone, and solute and vapor transport subsequent to immiscible plume establishment are then developed from the general model.

Water Resources Research

A compositional multiphase model for groundwater contamination by petroleum products: 2. Numerical solution

In this paper we develop a numerical solution to equations developed in part 1 (M. Y. Corapcioglu and A. L. Baehr, this issue) to predict the fate of an immiscible organic contaminant such as gasoline in the unsaturated zone subsequent to plume establishment. This solution, obtained by using a finite difference scheme and a method of forward projection to evaluate nonlinear coefficients, provides estimates of the flux of solubilized hydrocarbon constituents to groundwater from the portion of a spill which remains trapped in a soil after routine remedial efforts to recover the product have ceased. The procedure was used to solve the one-dimensional (vertical) form of the system of nonlinear partial differential equations defining the transport for each constituent of the product. Additionally, a homogeneous, isothermal soil with constant water content was assumed. An equilibrium assumption partitions the constituents between air, water, adsorbed, and immiscible phases. Free oxygen transport in the soil was also simulated to provide an upper bound estimate of aerobic biodgradation rates. Results are presented for a hypothetical gasoline consisting of eight groups of hydrocarbon constituents. Rates at which hydrocarbon mass is removed from the soil, entering either the atmosphere or groundwater, or is biodegraded are presented. A significant sensitivity to model parameters, particularly the parameters characterizing diffusive vapor transport, was discovered. We conclude that hydrocarbon solute composition in groundwater beneath a gasoline contaminated soil would be heavily weighted toward aromatic constituents like benzene, toluene, and xylene.

Water Resources Research

Behavior of sensitivities in the one-dimensional advection-dispersion equation: Implications for parameter estimation and sampling design

The spatial and temporal variability of sensitivities has a significant impact on parameter estimation and sampling design for studies of solute transport in porous media. Physical insight into the behavior of sensitivities is offered through an analysis of analytically derived sensitivities for the one-dimensional form of the advection-dispersion equation. When parameters are estimated in regression models of one-dimensional transport, the spatial and temporal variability in sensitivities influences variance and covariance of parameter estimates. Several principles account for the observed influence of sensitivities on parameter uncertainty. (1) Information about a physical parameter may be most accurately gained at points in space and time with a high sensitivity to the parameter. (2) As the distance of observation points from the upstream boundary increases, maximum sensitivity to velocity during passage of the solute front increases and the consequent estimate of velocity tends to have lower variance. (3) The frequency of sampling must be “in phase” with the S shape of the dispersion sensitivity curve to yield the most information on dispersion. (4) The sensitivity to the dispersion coefficient is usually at least an order of magnitude less than the sensitivity to velocity. (5) The assumed probability distribution of random error in observations of solute concentration determines the form of the sensitivities. (6) If variance in random error in observations is large, trends in sensitivities of observation points may be obscured by noise and thus have limited value in predicting variance in parameter estimates among designs. (7) Designs that minimize the variance of one parameter may not necessarily minimize the variance of other parameters. (8) The time and space interval over which an observation point is sensitive to a given parameter depends on the actual values of the parameters in the underlying physical system.

Water Resources Research