Neuman [1990] provided insight into a problem that has troubled groundwater hydrologists for at least the last ten years. He demonstrated that a fractal model appears to explain the scaling phenomenon of heterogeneity in aquifers, as measured by dispersivity values. The purpose of this comment is to put N euman's model into a geologic context. Neuman developed a universal scaling formula to relate dispersivity to the scale of observation but noted, "A major cause for local deviations from universal behavior appears to be the presence in many geologic media of discrete natural scales at which the log hydraulic conductivity field is homo-geneous" (p. 1756). These "discrete natural scales" are what geologists call facies. Specifically, a facies is a "three-dimensional body of rock having an environment origin that can be inferred from a set of characteristics including external geometry, internal geometry, sedimentary structures, lithology, organic content, stratigraphic relations, and associated facies" [Finley and Tyler, 1986, pp. 2-3]. Geologists apply the concept of facies on at least two scales: the local scale of Neuman's discrete natural scales (Figure 1 a) and a regional scale used in conceptual models of facies relationships (Figure lb). Following the dictates of a fractal model of heterogeneity, Neuman questioned the continued use of the representative elementary volume since "homogeneity is at best a local phenomenon limited to random and relatively narrow intervals of scale" (p. 1756). Yet, mapping geologic facies at a local scale may provide a way of defining meaningful representative elementary volumes. Miall [1985] extended the concept of facies by developing lithologic and geometrical descriptions of eight "architectural elements" that are found in alluvial systems. An element consists of a suite of facies that are associated with a specific depositional process. Examples include gravel bars and channel deposits. Davis et al. [1990] used Maill's architectural elements and facies descriptions when mapping an alluvial aquifer; they also measured permeability and calculated variograms for sections of the aquifer within an element. It is likely that such detailed geologic mapping is required to quantify heteroge-neity for accurate modeling of contaminant transport at certain scales. Given enough information on probable facies characteristics and distribution it may be possible to scale up heterogeneity and calculate effective parameters for contaminant transport modeling at other scales as is done in reservoir simulation [Lasseter et al., 1986]. It is important to note, however, that lumped measures of heterogeneity like effective parameters and Neuman's scaling rule cannot represent local peculiarities such as channel-ing of flow caused by the presence of connected units of high hydraulic conductivity. Indeed, Neuman cautioned that his scaling rule "does not necessarily describe conditions at any given locale but accounts for the self-similarity of log hydraulic conductivities in a mean sense over a la...

In this series of three papers a method is presented to estimate the parameters of groundwater flow models under steady and nonsteady state conditions. The parameters include values and directions of principal hydraulic conductivities (or transmissivities) in anisotropic media, specific storage (or storativity), interior and boundary recharge or leakage rates, coefficients of head-dependent interior and boundary sources, and boundary heads. In transient situations, the initial head distribution can also be estimated if the system is originally at a steady state. Paper 1 of the series discusses some of the advantage in treating the inverse problem statistically and in regularizing its solution by means of penalty criteria based on prior estimates of the parameters. The inverse problem is posed in the framework of maximum likelihood theory cast in a manner that accounts for prior information about the parameters. Since not all the factors which contribute to the prior errors can be quantified statistically at the outset, the covariance matrices of these errors are expressed in terms of several parameters which, if unknown, can be estimated jointly with the hydraulic parameters by a stagewise optimization process. When transient head data are separated by a fixed time interval, the temporal structure of these data is approximated by a lag-one autoregressive model with a correlation coefficient that can be treated as another unknown parameter. Estimation errors are analyzed by examining the lower bound of their covariance matrix in the eigenspace. Paper 1 concludes by suggesting that certain model identification criteria developed in the time series literature, all of which are based on the maximum likelihood concept, might be useful for selecting the best groundwater model (or the best method of parameterizing a particular model) among a number of given alternatives. if • = 0, prescribed head if •--• or, and a head-dependent flux (mixed) condition otherwise. When the flow is horizontal, K can be replaced by the transmissivity tensor, 'l', and Ss by the storativity, S. We will sometimes assume that the initial head, ho, satisfies a Poisson-type steady state equation V. (K. Vho) + qo = 0 on R (4) subject to -K. Vh 0 ß n = •(H0 -h) + Q on F (5)where q0, H0, and Q0 are the initial (steady state) equivalents of q, H, and Q. The solution of (1)-(5) requires a knowledge of the "aquifer parameters" K, Ss, q, H, Q, and • over the entire flow domain, R, and its boundary, F. In practice, (1)-(5) are often solved by numerical methods such as finite differences or finite elements in which the parameters take on discrete values. We will refer to such discrete values as "model parameters." While some aquifer parameters can be measured in the field, such measurements are usually scarce and prone to error. Furthermore, the measurements are generally performed on a scale different from that required for modeling purposes, so that the measured parameters are both conceptually and numerically different from their model counterparts. Si...

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A new analytical model is proposed for the delayed response process characterizing flow to a well in an unconfined aquifer. The present approach differs from that of Boulton [1954b, 1963, 1970] and Boulton and Pontin [1971] in that it is based only on well‐defined physical parameters of the aquifer system. Therefore it provides a possible physical explanation for the mechanism of delayed water table response and eliminates the conceptual difficulties encountered with Boulton's theory of ‘delayed yield from storage above the water table.’ Contrary to prevailing belief the process of delayed response in a homogeneous anisotropic phreatic aquifer can be simulated by using constant values of specific storage and specific yield without recourse to unsaturated flow theory. The results suggest that, in the absence of significant infiltration at the ground surface, compressibility may often be a much more important factor than unsaturated flow above the water table. Current methods of analyzing field data from unconfined aquifers do not usually consider compressibility. The present theory shows that such methods are limited in their application to relatively large values of time.

Hydrologic analyses typically rely on a single conceptual-mathematical model. Yet hydrologic environments are open and complex, rendering them prone to multiple interpretations and mathematical descriptions. Adopting only one of these may lead to statistical bias and underestimation of uncertainty. A comprehensive strategy for constructing alternative conceptual-mathematical models of subsurface flow and transport, selecting the best among them, and using them jointly to render optimum predictions under uncertainty has recently been developed by Neuman and Wierenga (2003). This paper describes a key formal element of this much broader and less formal strategy that concerns rendering optimum hydrologic predictions by means of several competing deterministic or stochastic models and assessing their joint predictive uncertainty. The paper proposes a Maximum Likelihood Bayesian Model Averaging (MLBMA) method to accomplish this goal. MLBMA incorporates both site characterization and site monitoring data so as to base the outcome on an optimum combination of prior information (scientific knowledge plus data) and model predictions. A preliminary example based on real data is included in the paper.

Previously, a new analytical model was proposed by the author for the delayed response process characterizing flow to a well in an unconfined aquifer. It was shown that contrary to prevailing belief this process can be simulated by using constant values of specific storage and specific yield without recourse to unsaturated flow theory. In the present work the theory is extended to account for the effect of a well partially penetrating a homogeneous anisotropic unconfined aquifer. Field and laboratory evidence is quoted to suggest that the elastic storage properties of unconfined aquifers may often be much more pronounced than those of deep‐seated confined aquifers composed of similar materials. The need for further research into the mechanical properties of unconfined aquifers and the role of unsaturated flow in the response process is emphasized.

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[1] Hydrologic systems are open and complex, rendering them prone to multiple conceptualizations and mathematical descriptions. There has been a growing tendency to postulate several alternative hydrologic models for a site and use model selection criteria to (1) rank these models, (2) eliminate some of them, and/or (3) weigh and average predictions and statistics generated by multiple models. This has led to some debate among hydrogeologists about the merits and demerits of common model selection (also known as model discrimination or information) criteria such as AIC, AICc, BIC, and KIC and some lack of clarity about the proper interpretation and mathematical representation of each criterion. We examine the model selection literature to find that (1) all published rigorous derivations of AIC and AICc require that the (true) model having generated the observational data be in the set of candidate models; (2) though BIC and KIC were originally derived by assuming that such a model is in the set, BIC has been rederived by Cavanaugh and Neath (1999) without the need for such an assumption; and (3) KIC reduces to BIC as the number of observations becomes large relative to the number of adjustable model parameters, implying that it likewise does not require the existence of a true model in the set of alternatives. We explain why KIC is the only criterion accounting validly for the likelihood of prior parameter estimates, elucidate the unique role that the Fisher information matrix plays in KIC, and demonstrate through an example that it imbues KIC with desirable model selection properties not shared by AIC, AICc, or BIC. Our example appears to provide the first comprehensive test of how AIC, AICc, BIC, and KIC weigh and rank alternative models in light of the models' predictive performance under cross validation with real hydrologic data.