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R.L. Naff

Publications and source records attributed to R.L. Naff.

11 recordsLinked to original sources
USGS2010

Arrival times and temporal moments of breakthrough curves for an imperfectly stratified aquifer

Using concepts developed in an earlier study, a solution in Laplace transform space is obtained for transport of resident concentration in an imperfectly but yet highly stratified porous medium. The flow field, into which an instantaneous pulse of tracer is injected, is taken to be steady and mean uniform parallel to the direction of stratification. From this transform-space solution either temporal moments can be derived by taking derivatives with respect to the Laplace parameter, or the transform-space solution can be inverted numerically to obtain breakthrough curves for the mean concentration. When compared to an equivalent solution with a Fickian dispersive flux, these temporal moments indicate the extent to which transport in heterogeneous porous media deviates from classical Fickian behavior. The numerical inversion of the Laplace transform solution gives partial breakthrough curves for the mean concentration which have the appearance of conflicting with the derived moment information. A hypothesis is put forth which resolves this apparent conflict; this hypothesis is verified by adding a component of local dispersion to the governing transport equation. On the basis of the flux-averaged concentration a form for the expected probability density function for the arrival time of a tracer particle is derived; arrival time moments and an arrival time cumulative distribution function are available as a consequence. Arrival time moments, as derived from the flux-averaged concentration, do not differ significantly from the resident moments, as derived from the resident concentration.

Water Resources Research↗
USGS2010

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↗
USGS2010

Depth-averaging effects on hydraulic head for media with stochastic hydraulic conductivity

Hydraulic conductivity of a porous medium frequently is considered to be a single realization of a three-dimensional spatial stochastic process. The most common observations of flow in porous media are hydraulic-head measurements obtained from wells which are screened over extensive sections of the medium. These measurements represent, approximately, a one-dimensional spatial average of the actual three-dimensional head distribution, the actual head distribution being a stochastic process resulting from flow through a random hydraulic-conductivity field. This paper examines, via ensemble averages, the effect of such spatial averages of groundwater flow on the spatial autocovariance function for a simple, yet viable, stochastic model of a bounded medium. The model is taken to be three-dimensional flow in a medium that is bounded above and below and in which the hydraulic conductivity is a second-order stationary stochastic process. Spatial averaging of the hydraulic heads is considered to take place over the entire thickness of the medium. Ensemble variances and autocorrelations for depth-averaged heads are computed for the resulting two-dimensional flow system and compared with those from a fully three-dimensional flow system.

Water Resources Research↗
USGS2010

A note on the recent natural gradient tracer test at the Borden Site

The variance in particle position, a measure of dispersion, is reviewed in the context of certain models of flow in random porous media. Asymptotic results for a highly stratified medium and an isotropic medium are particularly highlighted. Results of the natural gradient tracer test at the Borden site are reviewed in light of these models. This review suggests that the moments obtained for the conservative tracers at the Borden site could as well be explained by a model that more explicitly represents the three-dimensional nature of the flow field.

Borden Site↗
USGS2002

Shape functions for velocity interpolation in general hexahedral cells

Numerical methods for grids with irregular cells require discrete shape functions to approximate the distribution of quantities across cells. For control-volume mixed finite-element (CVMFE) methods, vector shape functions approximate velocities and vector test functions enforce a discrete form of Darcy's law. In this paper, a new vector shape function is developed for use with irregular, hexahedral cells (trilinear images of cubes). It interpolates velocities and fluxes quadratically, because as shown here, the usual Piola-transformed shape functions, which interpolate linearly, cannot match uniform flow on general hexahedral cells. Truncation-error estimates for the shape function are demonstrated. CVMFE simulations of uniform and non-uniform flow with irregular meshes show first- and second-order convergence of fluxes in the L2 norm in the presence and absence of singularities, respectively.

Computational Geosciences↗
USGS2000

Test functions for three-dimensional control-volume mixed finite-element methods on irregular grids

Numerical methods based on unstructured grids, with irregular cells, usually require discrete shape functions to approximate the distribution of quantities across cells. For control-volume mixed finite-element methods, vector shape functions are used to approximate the distribution of velocities across cells and vector test functions are used to minimize the error associated with the numerical approximation scheme. For a logically cubic mesh, the lowest-order shape functions are chosen in a natural way to conserve intercell fluxes that vary linearly in logical space. Vector test functions, while somewhat restricted by the mapping into the logical reference cube, admit a wider class of possibilities. Ideally, an error minimization procedure to select the test function from an acceptable class of candidates would be the best procedure. Lacking such a procedure, we first investigate the effect of possible test functions on the pressure distribution over the control volume; specifically, we look for test functions that allow for the elimination of intermediate pressures on cell faces. From these results, we select three forms for the test function for use in a control-volume mixed method code and subject them to an error analysis for different forms of grid irregularity; errors are reported in terms of the discrete L2 norm of the velocity error. Of these three forms, one appears to produce optimal results for most forms of grid irregularity.

Conference Paper↗
USGS1998

High-resolution Monte Carlo simulation of flow and conservative transport in heterogeneous porous media: 2. Transport results

In this, the second of two papers concerned with the use of numerical simulation to examine flow and transport parameters in heterogeneous porous media via Monte Carlo methods, results from the transport aspect of these simulations are reported on. Transport simulations contained herein assume a finite pulse input of conservative tracer, and the numerical technique endeavors to realistically simulate tracer spreading as the cloud moves through a heterogeneous medium. Medium heterogeneity is limited to the hydraulic conductivity field, and generation of this field assumes that the hydraulic-conductivity process is second-order stationary. Methods of estimating cloud moments, and the interpretation of these moments, are discussed. Techniques for estimation of large-time macrodispersivities from cloud second-moment data, and for the approximation of the standard errors associated with these macrodispersivities, are also presented. These moment and macrodispersivity estimation techniques were applied to tracer clouds resulting from transport scenarios generated by specific Monte Carlo simulations. Where feasible, moments and macrodispersivities resulting from the Monte Carlo simulations are compared with first- and second-order perturbation analyses. Some limited results concerning the possible ergodic nature of these simulations, and the presence of non-Gaussian behavior of the mean cloud, are reported on as well.

Water Resources Research↗
USGS1998

High-resolution Monte Carlo simulation of flow and conservative transport in heterogeneous porous media: 1. Methodology and flow results

In this, the first of two papers concerned with the use of numerical simulation to examine flow and transport parameters in heterogeneous porous media via Monte Carlo methods, various aspects of the modelling effort are examined. In particular, the need to save on core memory causes one to use only specific realizations that have certain initial characteristics; in effect, these transport simulations are conditioned by these characteristics. Also, the need to independently estimate length scales for the generated fields is discussed. The statistical uniformity of the flow field is investigated by plotting the variance of the seepage velocity for vector components in the x , y , and z directions. Finally, specific features of the velocity field itself are illuminated in this first paper. In particular, these data give one the opportunity to investigate the effective hydraulic conductivity in a flow field which is approximately statistically uniform; comparisons are made with first- and second-order perturbation analyses. The mean cloud velocity is examined to ascertain whether it is identical to the mean seepage velocity of the model. Finally, the variance in the cloud centroid velocity is examined for the effect of source size and differing strengths of local transverse dispersion.

Water Resources Research↗
USGS1998

A note on conservative transport in anisotropic, heterogeneous porous media in the presence of small-amplitude transients

The late-time macrodispersion coefficients are obtained for the case of flow in the presence of a small-scale deterministic transient in a three-dimensional anisotropic, heterogeneous medium. The transient is assumed to affect only the velocity component transverse to the mean flow direction and to take the form of a periodic function. For the case of a highly stratified medium, these late-time macrodispersion coefficients behave largely as the standard coefficients used in the transport equation. Only in the event that the medium is isotropic is it probable that significant deviations from the standard coefficients would occur.

Water Resources Research↗
USGS1985

Regression modeling of ground-water flow

Nonlinear multiple regression methods are developed to model and analyze groundwater flow systems. Complete descriptions of regression methodology as applied to groundwater flow models allow scientists and engineers engaged in flow modeling to apply the methods to a wide range of problems. Organization of the text proceeds from an introduction that discusses the general topic of groundwater flow modeling, to a review of basic statistics necessary to properly apply regression techniques, and then to the main topic: exposition and use of linear and nonlinear regression to model groundwater flow. Statistical procedures are given to analyze and use the regression models. A number of exercises and answers are included to exercise the student on nearly all the methods that are presented for modeling and statistical analysis. Three computer programs implement the more complex methods. These three are a general two-dimensional, steady-state regression model for flow in an anisotropic, heterogeneous porous medium, a program to calculate a measure of model nonlinearity with respect to the regression parameters, and a program to analyze model errors in computed dependent variables such as hydraulic head. (USGS)

Open-File Report↗