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A model for simulating microbial growth‐degradation processes in porous media is developed. It is assumed that the bulk of microorganisms in an aquifer grow in microcolonies attached to matrix surfaces. As developed, the model applies to the growth and decay of aerobic, heterotrophic microorganisms whose growth is limited by lack of a carbon and energy source (substrate), an oxygen source or both simultaneously as described by modified Monod kinetics. Transport of substrate and oxygen in the porous medium is assumed to be governed by advection‐dispersion equations with surface adsorption. A total of five coupled equations result describing substrate and oxygen concentrations in the pore fluid, substrate and oxygen concentrations in the microcolonies and colony density, which is assumed sufficiently small so that aquifer hydraulic conductivity is not diminished. An iterative process involving an Eulerian‐Lagrangian numerical procedure that is highly resistant to numerical dispersion in the presence of small dispersivities is used to solve the overall model, with parameter values selected from the literature or estimated. Results indicate that biodgradation would be expected to have a major effect on contaminant transport when proper conditions for growth exist. For one‐dimensional transport in a column, the most rapid microbial growth always occurred at the influent boundary where oxygen and substrate concentrations were held constant independent of colony density. Anaerobic conditions develop rapidly and aerobic biodegradation ceases if large amounts of substrate are added to the system.

Although the study of plants (botany) is one of the oldest sciences, relatively detailed quantitative theories of water transport in plant tissue have lagged behind those describing water transport in soils and other geologic materials which constitute the saturated and unsaturated zones. Many existing texts deal with various aspects of water transport in these earth materials, but little or nothing is devoted to the analogous transport of water in plant roots and tissue at a similar quantitative level. Yet the soil‐root‐stem water pathway is a major component of the subsurface hydrologic system. Evidently there is a need for both engineering and agricultural hydrologists to further develop their quantitative understanding of water movement in plant and soil‐plant systems. Modern quantitative theories of water transport in plants can be traced to concepts developed and disseminated effectively in landmark papers by Gradmann and van den Honert in 1928 and 1948 respectively. The material reviewed in this paper, while more advanced, is based on these concepts. Emphasis is placed on water movement in soil containing roots and on a general approach to water transport in living plant tissue. Detailed quantitative studies of water extraction by plant roots date back to studies by Gardner published in 1960. Many contemporary models are built around extraction functions in the Darcy‐Richards equation. Several such functions are listed in a table, and their applications, relative advantages, and limitations are discussed in the text. In a series of papers published in 1958, Philip developed the first detailed quantitative description of water transport in plant tissue. His approach resulted in a diffusion equation which could be written with water potential as the dependent variable. Philip's derivation assumed that water movement was primarily from vacuole to vacuole. Subsequent workers have refined and extended Philip's development to include water movement in cell walls and plasmodesmata. The development, interpretation, and application of these models over the past decade is presented in some detail. It can be argued that contemporary models of water transport in plant tissue are oversimplified. However, they have been subjected to some successful testing and they provide a framework within which to devise experiments. Moreover, the recent development of sophisticated experimental techniques should result in more detailed model testing during the 1980's.

Abstract. 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 L6vy motion and concepts of universal multifractals, which can be considered as generalizations of fBm/fGn, are reviewed briefly. IntroductionIn the hydrologic sciences, applications of fractional Brownian motion (fBm) and fractional Gaussian noise (fGn) evolved from the work of Hurst [1951, 1957] To the authors' knowledge the concepts and methodology described by Voss [1985aVoss [ , b, 1988 were first reviewed, integrated with contemporary stochastic and deterministic hydrologic concepts, and applied to transport processes in porous media by Hewett [1986]. The property that Hewett [1986] dealt with specifically was porosity. This was followed by applications to hydraulic conductivity [Moll and Bornan, 1993, 1995; Liu and Moll, 1996]. These references provide basic concepts and methodology for using fGn/fBm to represent heterogeneous hydraulic property distributions in porous media, along with supporting data. A recent related but independent application to the characterization of hydraulic conductivity at widely different scales of measurement was presented by Neurnan [1990Neurnan [ , 1994.The stochastic functions that we are calling fBm and fGn and various ways of generating such functions have been studied mathematically for a number of years [Mandelbrot and Van Ness, 1968;Mandelbrot, 1983]. In many applications the spectral properties of fBm/fGn become important but are no...

A knowledge of the variation of horizontal hydraulic conductivity with vertical position, K(z), is important in understanding the transport and dispersive properties of aquifers. Using an impeller meter to measure the discharge distribution in a screened well while pumping at a constant rate is a promising technique for obtaining the K(z) function. Such an application is described herein, and the resulting K(z) functions are compared with those obtained previously using tracer tests and multilevel slug tests. Impeller meter data were the most convenient to obtain, and tracer data the most difficult. The K(z) functions obtained by the three methods were not identical but quite similar overall. This similarity between both borehole tests and the larger-scale tracer test showed that nonstationary hydraulic conductivity trends, in a stochastic hydrologic sense, exist in the test aquifer. The impeller meter method was better able to detect the higher K layers than was the multilevel slug approach. Overall, the results suggest that a practical strategy for "fitting" impeller meter, tracer, or multilevel slug test data to a given aquifer is to use the selected testing procedure to obtain a dimensionless K/• distribution and then a standard pumping test to measure •. Combining both types of information enables dimensional values for K(z) to be calculated. In low permeability aquifers or near the bottom of a test well the fluid velocity due to pumping may be below the stall velocity of an impeller. Thus there is a definite need for the commercial development of more sensitive flow-measuring devices such as heat pulse flowmeters (Hess, 1986), which will extend the resolution of this field method. 1677