Fred J. Molz

A dual-domain mass transfer approach for modeling solute transport in heterogeneous aquifers: Application to the Macrodispersion Experiment (MADE) site

A large-scale natural-gradient tracer test in a highly heterogeneous aquifer at the Macrodispersion Experiment (MADE) site on the Columbus Air Force Base in Mississippi is simulated using three-dimensional hydraulic conductivity distributions derived from borehole flowmeter test data. Two methods of hydraulic conduct(fBm), are used to construct the hydraulic conductivity distributions needed by the numerical model. Calculated and observed mass distributions are compared to evaluate the effectiveness of the dual-domain mass transfer approach relative to the single-domain advection-dispersion approach. The results show that the classical Fickian advection-dispersion model can reproduce reasonably well the observed tritium plume above a certain concentration limit but fails to reproduce the extensive spreading of the tracer at diluted concentrations as observed in the field. The alternative dual-domain mass transfer model is able to represent the rapid, anomalous spreading significantly better while retaining high concentrations near the injection point. This study demonstrates that the dual-domain mass transfer approach may offer a practical solution to modeling solute transport in highly heterogeneous aquifers where small-scale preferential flow pathways cannot be fully and explicitly represented by the spatial discretization of the numerical model.

FRACTIONAL BROWNIAN MOTION AND FRACTIONAL GAUSSIAN NOISE IN SUBSURFACE HYDROLOGY : A REVIEW, PRESENTATION OF FUNDAMENTAL PROPERTIES, AND EXTENSIONS

Recent studies have shown that fractional Brownian motion (fBm) and fractional Gaussian noise (fGn) are useful in characterizing subsurface heterogeneities in addition to geophysical time series. Although these studies have led to a fairly good understanding of some aspects of fBm/fGn, a comprehensive introduction to these stochastic, fractal functions is still lacking in the subsurface hydrology literature. In this paper, efforts have been made to define fBm/fGn and present a development of their mathematical properties in a direct yet rigorous manner. Use of the spectral representation theorem allows one to derive spectral representations for fBm/fGn even though these functions do not have classical Fourier transforms. The discrete and truncated forms of these representations have served as a basis for synthetic generation of fBm/fGn. The discrete spectral representations are developed and various implications discussed. In particular, it is shown that a discrete form of the fBm spectral representation is equivalent to the well known Weierstrass-Mandelbrot random fractal function. Although the full implications are beyond the scope of the present paper, it is observed that discrete spectral representations of fBm constitute stationary processes even though fBm is nonstationary. A new and general spectral density function is introduced for construction of complicated, anisotropic, (3-D) fractals, including those characterized by vertical fGn and horizontal fBm. Such fractals are useful for modeling anisotropic subsurface heterogeneities but cannot be generated with existing schemes. Finally, some basic properties of fractional Levy motion and concepts of universal multifractals, which can be considered as generalizations of fBm/fGn, are reviewed briefly.

A FRACTAL-BASED STOCHASTIC INTERPOLATION SCHEME IN SUBSURFACE HYDROLOGY

Real porosity and hydraulic conductivity data do not vary smoothly over space, so an interpolation scheme that preserves irregularity is desirable. Such a scheme based on the properties of fractional Brownian motion (fBm) and fractional Gaussian noise (fGn) is presented. Following the methodology of Hewett (1986), the authors test for the presence of fGn in a set of 459 hydraulic conductivity (K) measurements. The use of rescaled-range analysis strongly indicated the presence of fGn when applied to the natural logs of the K data, and the resulting Hurst coefficient (H) was determined to be 0.82. This H value was then used along with the methodology for successive random additions to generate a fBm K interpolation (realization) in the vertical cross section between two wells. The results appeared realistic, and the overall methodology presented herein may serve as an improved basis for a conditional simulation approach to the study of various transport processes in porous media. (Copyright (c) 1993 American Geophysical Union.)

Multifractal analyses of hydraulic conductivity distributions

The concept of a universal multifractal, a generalization of a monofractal, is a recently developed scaling model for natural phenomena characterized by irregularity. Presented herein is an effort to use universal multifractal concepts to deal with spatial variations of hydraulic conductivity K, which have a significant effect on contaminant transport in the subsurface. Structure function analyses of four K data sets show that some vertical variations of K display multifractal structures, while others are consistent with monofractal behavior. In order to make multifractal concepts more useful, multifractal noise is introduced and defined as the increments of a multifractal. It is concluded that the multifractal formalism of Schertzer and Lovejoy [1987] has provided a rather general approach for modeling In K variations in the vertical. With the exception of horizontal variations of the Borden data, all results fell within the domain of universal multifractal behavior, which includes the monofractal case. Parameters were well-defined in an empirical sense and easy to calculate, indicating a robust formalism. Results were consistent with the recent finding of Liu and Molz [1997, also Non-Gaussian and scale-variant behavior in hydraulic conductivity distributions, submitted to Water Resources Research, 1997, hereinafter referred to as submitted paper] that K variations display increasing heterogeneity at decreasing scales.

Borehole flowmeters: field application and data analysis

Abstract This paper reviews application of borehole flowmeters in granular and fractured rocks. Basic data obtained in the field are the ambient flow log and the pumping-induced flow log. These basic logs may then be used to calculate other quantities of interest. The paper describes the applications of the Tennessee Valley Authority (TVA) electromagnetic (EM) flowmeter in a granular medium. In an attempt to ascertain the effect of formation disturbance on flowmeter readings, test wells were subjected to repeated periods of development using air lifting. Between the development periods, flowmeter tests were run. The results indicated that flowmeter data are not highly sensitive to formation disturbance in the near vicinity of a test well. Evidence of changes in the natural (ambient) flow in the test well was attributed to high, variable rates of pumping by nearby chemical plants. In fractured media, flowmeters may be used to detect flow to or from individual fractures or fracture zones. In these applications, additional information in the form of caliper, televiewer and resistivity logs are useful. Further studies of the type described herein are warranted.

A physically based, two-dimensional, finite-difference algorithm for modeling variably saturated flow

Abstract A computationally simple, numerical algorithm capable of solving a wide variety of two-dimensional, variably saturated flow problems is developed. Recent advances in modeling variably saturated flow are incorporated into the algorithm. A physically based form of the general, variably saturated flow equation is solved using finite differences (centered in space, fully implicit in time) employing the modified Picard iteration scheme to determine the temporal derivative of the water content. The algorithm avoids mass-balance errors in unsaturated regions and is numerically stable. The resulting system of linear equations is solved by a preconditioned conjugate gradient method, which is known to be computationally efficient for the type of equation set obtained. The algorithm is presented in sufficient detail to allow others to implement it easily, and is verified using four published, illustrative sets of experimental data.

Further evidence of fractal structure in hydraulic conductivity distributions

Past rescaled range analyses of porosity and hydraulic conductivity (K) distributions have indicated the presence of long-range correlations in the data typical of the related stochastic functions known as fractional Gaussian noise and fractional Brownian motion [Hewett, 1986; Molz and Boman, 1993]. New K data analyzed herein lend further support to this notion. Horizontal processes that mimic fBm will display a power-law variogram. The Mandelbrot-Weierstrass random fractal function is introduced as an analytical model for fBm and used to illustrate several concepts. With the exception of the Hurst coefficient value (H), our analysis supports the existence of fractal-like K fields similar to those visualized by Neuman [1990, 1994]. Past studies which produced H values greater than or less than 0.5 appear to differ mainly because of the underlying model, fractional motion or fractional noise, that was assumed in the respective analyses.

A 2-D, diffusion-based, wetland flow model

Surface water flow in wetlands occurs typically in a low gradient environment where topographic and flow resistance variations control the discharge distribution. It is within this fluid mechanical regime that the biochemical processes constituting wetland ecology take place. Presented herein are the basics of a mathematical model that takes advantage of the hydraulics typical of wetlands. The resulting model, in the form of a 2-D, non-linear diffusion equation, allows incorporation of spatial variations in flow resistance and topography. The applicability of the model to one- and two-dimensional wetland type flow is demonstrated using two cases: a laboratory experiment and a wetland pond. Results illustrate the versatility of the formulation and the important influence of topography variations on flow depth and velocity variations. The use of a fixed rectangular calculation domain to simulate the flow pattern in wetlands with irregular wet boundaries is the most significant practical aspect of the WETFLOW model. Future research will concentrate on dealing with extreme heterogeneity in the elevation and flow resistance distributions, through a combination of measurements and use of stochastic fractals to represent property distributions (Molz and Boman, 1993, Water Resour. Res., 29: 3769–3774). Experimental studies are needed to quantify the relationship of flow resistance to different combinations of slope, water depth, and a suitable measure of vegetation density for a wetland ecosystem.

Fractional Laplace model for hydraulic conductivity

[1] Based on an examination of K data from four different sites, a new stochastic fractal model, fractional Laplace motion, is proposed. This model is based on the assumption of spatially stationary ln(K) increments governed by the Laplace PDF, with the increments named fractional Laplace noise. Similar behavior has been reported for other increment processes (often called fluctuations) in the fields of finance and turbulence. The Laplace PDF serves as the basis for a stochastic fractal as a result of the geometric central limit theorem. All Laplace processes reduce to their Gaussian analogs for sufficiently large lags, which may explain the apparent contradiction between large-scale models based on fractional Brownian motion and non-Gaussian behavior on smaller scales.

Water transport through plant tissue: the apoplasm and symplasm pathways

Abstract A detailed quantitative analysis of water flow through the apoplasm and symplasm of plant tissue is presented. The analysis results in two coupled diffusion equations which describe water transport in the two pathways. Various parameters entering the analysis identify the physical properties of the tissue which control the transport process as the resistance to water flow per cell in the two parallel pathways, the resistance per cell between pathways, and the water capacity per cell in the two pathways. Values for the several resistances and water capacities are estimated from available data, and a model problem is solved wherein a sheet of tissue at an initial water potential of — δ bars is immersed in a container of water. The resulting solutions show that depending on the values assigned to the controlling parameters, local water potential equilibrium between each cell and its cell wall may or may not obtain. In the special case of local equilibrium (water potential in the symplasm and apoplasm pathways essentially equal), the transport process can be described by a single diffusion equation which is derived along with an expression for the tissue diffusivity. It is concluded that the weakest link in the analysis is the estimated value for the permeability of the plasmodesma membrane, and that a logical extension of the theory would be to include the effects of a diffusable solute.