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The paper, 'Drawdown in a Well of Large Diameter,' as the authors themselves have pointed out [ Papadopulos and Cooper, 1967], has been written with a view to help hydrogeologists in Asian countries, where ground water is developed to a large extent by means of large diameter open wells. The method suggested in the paper was applied by this writer to analyze drawdown data from a few large diameter wells piercing alluvial sands, and satisfactory results were obtained. Specifically, the method proposed is valid for aquifers with relatively high transmissibilities or wells of relatively small diameters. That is, the method is most satisfactorily applied in those cases in which the quantity of water developed from storage in the open well is small compared with that developed from inflow into the well. It has been pointed out [ Papadopulos and Cooper, 1967, p. 244], 'Therefore, data that completely fall on the almost straight line part of the type curves cannot be analyzed by this method, or, in fact, by any other method.' [ABSTRACT FROM AUTHOR]

The title of this very interesting paper [ Kirkham, 1967] provides a clue to what we believe to be a misconception that has led the author to his analysis, which is in itself impeccable. It is the essential nature of an approximation that it assumes certain things that are not strictly true and that are known not to be true; otherwise it would not be an approximation but a rigorous treatment. If an admitted approximation, such as the Dupuit-Forchheimer treatment, is then submitted to rigorous analysis as though no approximation had been made, then naturally some nonsensical results will emerge, as Muskat showed in the passage referred to by Professor Kirkham. To that extent every approximate treatment involves a paradox and is accepted as involving a paradox. If, and only if, the assumption is not too far from the truth, do the results calculated from it also prove to be sufficiently near the mark, and then, and only then, is the approximation useful. [ABSTRACT FROM AUTHOR]

The disposal of waste water into a confined stratum of the earth's crust through recharge wells is gaining increasing popularity. Underground burying of radioactive liquid and solid wastes has been practiced for years at Hanford, Oakridge, and other places. These highly dangerous wastes can diffuse out in lateral directions if the containers crack due to earthquakes and internal pressures in relatively thin confined stratum. These situations are often characterized as the absorption of moisture by the partially saturated soil surrounding the disposal site. This phenomenon is governed by a nonlinear partial differential equation, subject to initial and boundary conditions. A method of weighted residuals is applied to extract an explicit solution, which is simple for applications. The solution is valid for any form of the diffusivity function and can be used with any value of initial moisture content. The problem with the initial soil moisture at the threshold level (the minimum moisture content at which the permeability exists) has been investigated by Philip, and the corresponding results of the present paper compare very well with the results obtained by Philip. [ABSTRACT FROM AUTHOR]

This paper is a continuation of a previous one in which a stochastic model for analyzing excessive streamflows was presented. The model was based on the assumption that flood peak exceedances are independent, identically distributed random variables and that their occurrence is subject to the Poisson law. In the present paper a model is developed for nonidentically distributed exceedances on the assumption that only those exceedances during a particular season may be considered identically distributed. From this hypothesis the distribution function of the maximum flood peak exceedance in an arbitrary interval of time (0, t] is determined. The results are then applied to the 72-year record of the Greenbrier River at Alderson, West Virginia. The theoretical and observed results agree reasonably well. [ABSTRACT FROM AUTHOR]

The 3-parameter lognormal distribution is a general skew distribution in which the logarithm of any linear function of a given variable is normally distributed. The distribution is applied to the frequency analysis of floods, annual flows, and monthly flows, and a comparison with other commonly used methods suggests that it can be successfully used for this purpose. A procedure for its application has been suggested using only the median, the mean, and the standard deviation of the original data. The Gumbel distribution is a special case, and any straight line on the Gumbel probability paper can be transformed into a straight line on the lognormal probability paper by the 3-parameter lognormal distribution. The sample of 10 stations used in this study all exhibited negative skewness for the logarithms of the data and therefore the lognormal distribution that assumes this skewness equal to zero has generally given high estimates. [ABSTRACT FROM AUTHOR]

This paper presents a minimum cost method to time and size water system component expansion to meet an exogenously determined growth rate. The method analyzes the economic impact both of the economies of scale in construction and of the real cost of capital. The result is a capacity expansion model which determines when and how much excess capacity should be installed to meet increasing demands. The economic effects of timing and sizing decisions on total costs are plotted for various interest rates and levels of economies of scale. The paper demonstrates that the costs of expansion are sensitive to the increase in demand specific to the facility being planned and that the design capacity must be set individually for each separable water system component according to its particular cost function. These findings suggest that the standard practice of designing excess capacity in a water system to meet a fixed future demand (i.e., 25-year system) should be revised. [ABSTRACT FROM AUTHOR]

Conventionally hydrologists have used regression analysis for solving their multivariate problems. Recently other multivariate statistical methods have been advocated. This paper discusses and compares the effectiveness of six methods of analysis: regression, principal component, varimax, oblimax, key cluster, and object. Strengths and weaknesses of each method are discussed, and the combination of principal component regression with varimax rotation of the factor weight matrix is recommended for an initial analysis of multifactor hydrologic problems. [ABSTRACT FROM AUTHOR]

The dynamics of leaky aquifers are governed by a system of integrodifferential equations derived in this paper. Alternative expressions for the memory functions are obtained, and it is shown that approximate theories of leaky aquifers correspond to several ways of approximating the memory functions. [ABSTRACT FROM AUTHOR]

This paper discusses some new water allocation policies that maximize expected profit when the central authority controlling a reservoir faces deterministic demand functions and a stochastic supply of water. The demand functions are distinguished by price and a guaranteed probability of receiving water. A penalty cost is incurred if the guaranteed probability of delivery is not met by the reservoir manager. The demand functions could represent, for example, consumers of water for the purposes of hydroelectric power, recreation, irrigation, and dilution for pollution control. The optimal contract policies (optimal prices and associated optimal guaranteed delivery probabilities) are characterized. The analysis shows that the use of price discrimination may be superior (Pareto's law) to a uniform contract for all consumers of water. The results are illustrated by a stochastic model of two classes of water consumers. [ABSTRACT FROM AUTHOR]

In the calibration of aquifer models, the desire for an automated adjustment process is sometimes in conflict with the need for subjective intervention during the calibration process. Working from the established concept of modeling for unconfined aquifers with rectangular nodes, this paper describes a method of obtaining subjective information in addition to that customarily used in model calibration. Such information is assembled and quantified at the same time as the usual data are gathered for initial estimation of parameters. This information is then introduced into an automatic adjustment process in such a way that the calibration process can proceed without interruption until the desired level of relative accuracy is reached, or until it is shown that the desired level cannot be reached within the constraints assigned. A decoupling technique that permits (under restricted conditions) simultaneous adjustment of all parameters is also described. A digital computer model of the Tucson basin aquifer is used to illustrate the concepts and demonstrate the adjustment method. [ABSTRACT FROM AUTHOR]

A series of models has been developed in previous papers to estimate the best sized area to develop for irrigation from a given reservoir under conditions of variable supply and demand for irrigation water. The best operating policies for such a system were also derived. The approach of this paper extends the previous approach by recognizing the value of water carried over from one season to the next. A modified version of the previous approach is used to generate reservoir level transition probabilities and expected benefits. A dynamic programing model is then used to estimate the expected benefits from allocating water optimally between seasons. These expected benefits are a function of acreage developed for irrigation and beginning-season reservoir levels. Such a function allows a decision maker at any decision point within an irrigation season to equate the expected net benefits from allocating water to the rest of the current season with the expected net benefits from saving such water for the following season. This decision-making process is simulated for a number of seasons. The results of an application of the models indicate that such interseasonal transfer can considerably increase the present value of expected benefits from the project. [ABSTRACT FROM AUTHOR]

This paper deals with floodplain land use management approaches for urban flood damage control. These alternatives are important because traditional flood control projects, such as levees, channel improvements, and reservoirs, are not always capable of correcting the physical and economic conditions that give rise to flood damage. It is expected that land use management will complement engineering works and lead to more effective use and development of floodplain lands. The paper presents a computational technique for evaluating alternative land use assignments based upon the economic value a community gains from its land. A linear programing model is developed that identifies economically efficient combinations of (1) spatial and temporal planning of urban land use, (2) site elevation through landfill, and (3) flood proofing of buildings. [ABSTRACT FROM AUTHOR]

This paper discusses the problems involved in identifying valid demand and supply parameters estimated from sample data on recreation. It is argued that an analysis of cross section data permits the quantification of the effects of demand variables. Cross section results can then be pooled with recreation time series to enable the valid estimation of supply parameters. These supply parameters can be used to improve decisions regarding the expansion and development of recreation sites. More refined projections of rates of use can be derived for use as benefit measures in cost-benefit analysis. An example of the pooling technique is presented using demand equations estimated from the National Recreation Surveys and a time series of visitation data at the Tennessee Valley lakes. The results tend to confirm the superiority of the method proposed, particularly in comparison with an analysis using only time series data. [ABSTRACT FROM AUTHOR]

In a study of the probability of exceedance of a selected stage for daily maximum water levels, exponential and gamma models can be shown to be more realistic than the Markov model described in the first paper of this series. The important differences are in the tails of the distributions, and the estimate of the expected recurrence interval is affected little by the choice of model. Confidence limits for the mean recurrence interval are, in fact, somewhat wider for the exponential model than for the Markov model. It is shown that the general relationship between the expected recurrence interval and the return period for the annual maximum series depends on the choice of model. [ABSTRACT FROM AUTHOR]

The statistical theory of meander formation ascribes the meandering of stream channels to the hypothesis that a river has a Gaussian probability to deviate by an angle dΦ in progressing an elemental distance ds along its course. This paper aims at checking this hypothesis. Taking three cases, the Salt Fork Vermilion River in Illinois, the Assiniboine River in Canada, and, as an interesting comparison, a 'river' (Hadley Rill) on the Moon, it was indeed possible to show that the probability in question is Gaussian, using pertinent statistical tests. Our study can therefore be said to confirm to this extent the probabilistic view of meander development. [ABSTRACT FROM AUTHOR]

Recent years have seen a marked increase in the rate of investment in facilities for water quality control and a concurrent recognition of the need to program water resource use on a regional basis. This paper outlines the nature of regional water quality systems and presents programming models for several water quality improvement problems. A linear programming formulation of a static improvement in quality is contrasted with a traditional policy for programming improvements; the traditional policy results in a mixed integer problem. The maximization of the ratio of benefits to costs of an improvement program is transformed to a linear programming problem. A stochastic natural environment leads to other programming models. A discussion of the application of least-cost models terminates the paper. [ABSTRACT FROM AUTHOR]

We appreciate the interest shown in our paper [Meier and Beightler, 1967] by Brater in his comment. Brater raises two significant and valid points concerning the paper. First and foremost, there is a numerical error in the authors' paper, as indicated in a previous reply [Meier and Beightler, 1968]. The correct values of d1 and d2 in Equation 16 of the authors' paper should be 7.11 and 2.11, respectively. Furthermore, the total maximum return is slightly greater than 355. These are typographical errors and were not occasioned through the use of the particular optimization method employed. [ABSTRACT FROM AUTHOR]

The Research Committee on Water Resources Systems is the youngest in the family of committees of the Section on Hydrology. Since its organization in 1966, the committee has set itself not only the goal of furthering research in the analysis of water resources systems, but of bringing together researchers and practitioners in a meaningful exchange of ideas. The symposium presented on April 12, 1971, at the 52nd Annual Meeting of the American Geophysical Union in Washington, D.C., was the first effort of the committee to present a program of invited papers. [ABSTRACT FROM AUTHOR]

The authors wish to express their appreciation to Loucks [1968] for his constructive criticism of their recent paper [Meier and Beightler, 1967]. Loucks' comment correctly interprets the authors' enthusiasm for dynamic programming and its usefulness in water resource analysis. The authors believe that dynamic programming has an important role to play in the analysis of the large-scale system planning problems occurring in the water resources area. [ABSTRACT FROM AUTHOR]

This paper first examines three methods currently used for apportioning the costs of joint waste water treatment facilities among users. The methods are cost sharing based on the measure of pollution, cost sharing based on single plant costs with a rebate proportional to the measure of pollution, and cost sharing based on the separable costs remaining benefit method. Through the use of numerical and theoretical arguments it is demonstrated that often these methods do not provide an apportionment that satisfies all the people involved and that a potential participant may find it economically advantageous not to join the regional plan and may thereby force the adoption of a system that is, in total, more costly. These difficulties will frequently arise when any plan is subjected to the pressures of 'free market' bargaining. Two additional methods relying on game-theoretic concepts are then introduced to deal with the problem. These methods, which are practical, can satisfactorily solve problems that are not tractable when other techniques are used. Also, the basically perverse characteristics of situations requiring cost sharing are demonstrated by using the bargaining methods to prove that some cost-sharing problems have no unique solutions and that others have no viable solution at all unless the regional authority can obtain a subsidy. [ABSTRACT FROM AUTHOR]

Modification of watersheds occurs either through natural processes, such as erosion, or human influences, such as urbanization. In either case the rainfall input must be properly modeled before the runoff output can be predicted as the modifications take place. The paper considers runoff-producing summer precipitation of short duration and high spatial variability as an intermittent stochastic phenomenon. The probability distribution of seasonal total point or areal rainfall is obtained by convoluting a Poisson number of events with a geometric or negative binomial probability of rainfall amount. Close agreement with the experimental data is found. Next the probability of various combinations of rainfall amounts, given the seasonal total and the number of events, is computed. With these results, the theoretical seasonal water yield distribution can be obtained by using a simple rainfall-runoff relationship, such as the Soil Conservation Service formula. The possibility of using regional input parameters to study the distribution of the output of poorly gaged small watersheds is discussed. In particular, extreme total flows can be computed. [ABSTRACT FROM AUTHOR]

This paper presents an integrated analysis of flow through earth dams in which the entire porous domain is considered. The mathematical model provides a finite difference solution to two- or three-dimensional problems involving transient or steady state flow in the saturated and unsaturated domains of nonhomogeneous, anisotropic dam sections. In general, the water table is not a streamline; in many zoned cross sections, a significant proportion of the stream tubes may take unsaturated flow routes for some part of their traverse. In addition, the position of the water table may be quite different from that determined from the classic saturated-only analysis. Unsaturated flow components take on greater importance in small dams rather than in large, in dams with sloping cores and downstream filter blankets rather than more homogeneous sections, and in the fine-grained, well-graded soils common to internal cores rather than in the more permeable and more uniform soils of the external sections. A simulation of the transient initial advance problem shows that the rate of growth of the saturated zone is highly dependent on the unsaturated properties of the soils and on the initial moisture contents. In the early stages of seepage following a rapid rise in the upstream reservoir level, inflow rates into the dam may be as much as 100 times the steady state seepage rate. Consideration of the unsaturated zone in the analysis of seepage through earth dams involves added mathematical complexity and requires data on unsaturated soil properties that are not commonly available. [ABSTRACT FROM AUTHOR]

A technique is described for solving various problems in continuum mechanics through the application of similarity transformations when all the requirements for the application of such transformations are not satisfied. The transformed equations usually involve a stray time variable that is interpreted as a system parameter. When this parameter is properly interpreted, the solutions of the transformed systems are very good numerical approximations for the solutions of the original systems. The advantages of considering the transformed systems are usually two-fold: numerical solutions can be obtained with less computer time, and qualitative analysis of the transformed systems is usually easier than that of the original systems. The problems discussed in this paper include certain previously published results and some new results obtained for the nonlinear Fokker-Planck equation in which the vertical movement of soil moisture under the influence of gravity is described. [ABSTRACT FROM AUTHOR]

Three problems involved in planning and operating an irrigation system are (1) the short-run problem of allocating a quantity of water in time over an irrigation season, (2) the intermediate-run problem of deciding what area of crop to plant at the beginning of an irrigation season, and (3) the long-run problem of deciding what area of land to develop for irrigation. This paper concerns the intermediate-run problem. Crop growth is simulated with a simple crop growth model over a number of years on a large acreage supplied with water from a reservoir subject to stochastic inflows; the demand for water by the crop also being stochastic. Irrigation decision rules at each of a number of decision points in time are taken from the output of a simulation-dynamic programing model used to solve the short-run problem. The approach used allows for the abandonment for the rest of the season of portions of the area initially irrigated but does not allow for temporary abandonment, and reinstatement during the season, which is a limitation. The results show the best irrigated acreage to plant to be an approximately linear function of the reservoir contents at the beginning of the season. The function is found to be highly sensitive to changes in the profitability of the best dryland alternative enterprise. The cost of planting suboptimal acreages is found to be high. [ABSTRACT FROM AUTHOR]

Baseflow as a hydrograph component is discussed with special reference to Hall's [1968] paper 'Base-Flow Recession-A Review'. Additional references with particular emphasis on the baseflow problem are given. A three parameter baseflow partition functional is shown to give substance to the concept of 'group persistence' and to suggest an underlying mechanism to the Hurst statistical phenomenon. [ABSTRACT FROM AUTHOR]

The present study follows two papers [ De Geoffroy et al., 1967, 1968] that describe a spring sampling technique used for the geochemical coverage of a large area of southwest Wisconsin to detect zinc ore bodies. During the course of the geochemical survey, it became apparent that valuable hydrologie data could be collected simultaneously for a small additional cost. Little information on springs is available in the literature for Wisconsin and most parts of the United States, with the exception of the southwest portion of the country [ Feth, 1953; Feth et al, 1964; Waring, 1915]. It is hoped that the present survey of springs can be a useful addition to the study of ground water in southwest Wisconsin. Samples from 7210 springs were collected within an area of 900 square miles (Figure 1) during four field seasons. The systematic procedure covering the field work has been detailed previously [ De Geoffroy et al., 1967]. [ABSTRACT FROM AUTHOR]

Although the problems of water quality management are usually analyzed in the context of nonstochastic models, such an analysis may be misleading because of the substantial stochastic elements in streamflow. The effects of variance in streamflow are considered in this paper. The analysis shows that the problems of uncertainty may be dealt with by selecting a critical value of streamflow ƒθ and then by treating pollutants sufficiently so that if streamflow falls above (or below) that critical value the water quality standard will (or will not) be met. The greater the variance in streamflow, the lower the optimal critical value. Further, the level of treatment of pollutants is higher under uncertainty than it would be if there were no variance in streamflow. [ABSTRACT FROM AUTHOR]

In this paper we have two main objectives. The first of these is to outline a research and payoff (R&P) model that suggests a procedure for evaluating competing or concurrent research projects. The atmospheric sciences provide a good example for an initial application of the model. Weather modification, particularly artificial stimulation of precipitation, has been touted as an answer to many water resource problems. Alternatives to this technology have received little attention. We suggest that the R&P model could profitably be applied in this instance. The second objective of the paper is incorporated in a pilot study of the first step in such an application: namely, the determination of the estimated probabilities of success of various technologies in the atmospheric sciences. For this step, responses to a questionnaire sent to professional atmospheric scientists were examined to determine whether (a) scientists are willing to provide probability estimates of success of various technologies and whether these estimates are consistent with other estimates of success; (b) personal involvement in a field influences such estimates; (c) the probability estimates are consistent with the respondents' assignment of shares of a research budget; and (d) the estimates are generally in agreement with the reports of 'blue-ribbon' panels of experts. We conclude that it is possible to obtain reasonably consistent estimates of research success from those who know most about them. The degree of optimism among those sampled is high, and it is highest for those workers involved in and having a high degree of knowledge of a particular field. Continued efforts in the determination of research success probabilities specifically tailored to the R&P model are recommended. [ABSTRACT FROM AUTHOR]

An integrated system for collecting, recording, and analyzing data describing the behavior of water flowing through a forest soil has been developed. The collecting and recording components are located on a lower terrace of the City of Seattle Cedar River watershed in western Washington. This system includes tension lysimeters installed such that the overlying soil and rooting systems are undisturbed. The lysimeters are constructed from a fused aluminum oxide disk 11 inches (28 cm) in diameter. The suction to each lysimeter is maintained through a single controlled vacuum source. Solutions collected during periods of flow are passed through flow cells measuring conductivity, acidity, and rate of water flow. The sensitivity of the soil moisture flowmeter is in excess of 0.002 cm/hr. The electrical output from the flow cells is coupled to a data-logging facility and printed as perforations in paper tape. Computer analysis converts these data to tabular and graphic form. Examples demonstrating the functioning of the system during periods of flow are presented. [ABSTRACT FROM AUTHOR]

An exact solution is derived for the second term of J. R. Philip's [1957] series expansion originally developed for his numerical method of solution of the diffusion equation with diffusivity concentration dependent. The exact solution is made possible by the adoption of simple algebraic functions to describe the water content-capillary suction and the water content-capillary conductivity relationships of the porous medium and of a solution for horizontal infiltration presented in an earlier paper [ Brutsaert, 1968]. Because of the fast convergence of the series, all terms of order higher than the second are negligible. Thus the present solution for the second term used in combination with the first term, representing the solution for horizontal infiltration, provides an adequate solution for vertical infiltration. Good agreement is obtained with published experimental data. [ABSTRACT FROM AUTHOR]

By 'Noah Effect' we designate the observation that extreme precipitation can be very extreme indeed, and by 'Joseph Effect' the finding that a long period of unusual (high or low) precipitation can be extremely long. Current models of statistical hydrology cannot account for either effect and must be superseded. As a replacement, 'self-similar' models appear very promising. They account particularly well for the remarkable empirical observations of Harold Edwin Hurst. The present paper introduces and summarizes a series of investigations on self-similar operational hydrology. [ABSTRACT FROM AUTHOR]

When two miscible liquids flow through a porous medium (e.g., fresh water and salt water in an aquifer), the dispersion at the interface is described by Fick's law with a velocity-dependent coefficient. In the first part of this paper, the horizontal flow of two liquids is used to demonstrate that the influence of gravity and the effects of the differences in density and viscosity are usually negligible. Then solutions are presented for the dispersion at the interface of liquids in two-dimensional flow. [ABSTRACT FROM AUTHOR]

From analysis of the water profiles obtained by gamma ray absorption during infiltration into soil columns, and of the wetting branch of the soil water-suction water content relation, this paper suggests a method for checking the validity of the generalized Darcy's law and for computing the values of the hydraulic conductivity at various water contents. Tests have been made for horizontal infiltration and for capillary rise. In both tests, for a given water content, the capillary conductivity k is constant, and the generalized Darcy's law is valid and may be used to describe a transient flow in unsaturated soils. [ABSTRACT FROM AUTHOR]

Prediction equations for hydrologic events developed from one population of observations (watersheds) are often solved for another population that is removed either in time or in space. Predictions of this kind are never certainties, although some predictions are obviously more uncertain than others. This paper proposes an empirical uncertainty classification that may be found useful for separating probably successful from probably unsuccessful extensions of prediction equations. The classification system is illustrated by a prediction equation for suspended sediment discharge developed from some watersheds in California, and by a discriminant function for marine versus nonmarine sediments based upon microelements. (Key words: Statistics; principal components analysis; linear discriminant functions). [ABSTRACT FROM AUTHOR]