# Regression Modeling of Ground-Water Flow

### U.S. Geological Survey, Techniques of Water-Resources Investigations, Book 3, Chapter B4

By Richard L. Cooley and Richard L. Naff

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## Preface

Scientists and engineers have been using ground-water flow models to study ground-water flow systems for more than 20 years. The basic modeling process seems to be relatively straightforward. Initially, a sound conceptual model is formed and is translated into a tractable, mathematical model. Contributing to (and following) this conceptualization process is the collection of field information, such as (1) location and extent of hydrostratigraphic units, recharge areas, discharge areas, and system boundaries; (2) hydraulic head measurements; and (3) pumping discharges. These data form the basis for input to the flow model. Finally, the model is run, and the desired information such as head distribution or flux rates is extracted. However, people engaged in modeling usually observe that two pervasive problems considerably complicate the situation. One problem is that good, general methods of measuring (or computing) some of the variables that characterize the flow system and its geologic framework do not exist. One example is measurement of ground-water recharge. No direct ways of measuring recharge exist, and the accuracy of indirect methods is often unknown. Furthermore, many indirect methods are applicable only to unique situations. The second problem relates to errors in the measurements and their propagation into model results. No error-free measurement (or computation) methods for obtaining data on the flow system exist. Thus, even the variables that can be estimated will contribute to error, so that model results will always be unreliable to some extent. As a consequence of these two problems, measurement (or computation) of the necessary input variables, application of them to an adequate model, and calculation of the desired results to an acceptable accuracy generally are not possible. Other methods that recognize and deal with the problems of incompleteness and (or) inaccuracy of data must also be applied. The present text has been designed to teach these methods to scientists and engineers engaged in ground-water modeling.

The basic methodology is multiple, nonlinear regression, in which the regression model is some type of ground-water flow model. As seen subsequently, this methodology is consistent with known aspects of the physical systems to be analyzed and requires relatively few assumptions. Even though the present text is directed specifically toward ground-water modeling, the procedures to be discussed are applicable to a number of different types of modeling problems. Thus, the methods are usually discussed in a general context; in other words, without reference to any specific model.

Material in the present text evolved from notes developed for training courses in parameter estimation for ground-water flow models taught by the authors and others at the U.S. Geological Survey National Training Center, Denver Federal Center, Lakewood, Colo. The philosophy of these courses, and of this text, is to teach general methods that are applicable to a wide range of problems and to teach these methods in sufficient depth so that students can apply them to many problem situations not considered in the courses or text.

The main body of the text is organized into six major sections. The first section is an introduction that discusses the general topic of modeling ground-water flow. This section shows that ground-water modeling problems are an incomplete combination of direct-type problems (solution for hydraulic head given values of flow system and framework variables) and inverse-type problems (solution for flow system and framework variables given values of hydraulic head) that commonly require solution by optimization procedures which give the best fit between observed and calculated results. Because the specific optimization approach employed here is regression and regression procedures are based on statistical concepts, the second section is included to provide the student with the necessary statistical background material. It is not designed to be an exhaustive review of basic statistics; rather, it presents material essential to understanding the following sections. The third section presents detailed material on linear and nonlinear regression. Although most of the material on linear regression is fairly standard, some of the material on nonlinear regression is not. In particular, specific modifications presented to induce convergence of the iterative solution procedure for nonlinear regression have not, to the writers’ knowledge, been presented elsewhere in the form given here. The fourth section applies the nonlinear regression method to the specific problem of developing a general finite difference model of steady-state ground-water flow. In the fifth section, statistical procedures are given to analyze and use general linear and nonlinear regression models. The tests and analytical procedures presented are not exhaustive; they are the ones that the writers have found to be most useful for analyzing the real systems examined to date The sixth section is designed to be supplemental to the preceding sections. Specialized procedures presented include nonlinear regression for models that cannot be solved directly for the dependent variable, a measure of model nonlinearity called Beale’s measure, and a statistical test for compatibility of prior information on parameters and parameter estimates derived from sample (observed head) information.

A number of exercises have been included, and a complete discussion of the answers can be found in the seventh major section at the end of the text. These problems exercise the student on nearly all methods presented. In addition, three computer programs are documented and listed: the program for nonlinear-regression solution of ground-water flow problems of section four, a program to calculate Beale’s measure, and a program to calculate simulated errors in computed dependent variables such as hydraulic head.

The mathematical background necessary to use this text includes basic mathematics through differential and integral calculus, including partial derivatives, and matrix algebra. A background in elementary statistics would be useful but is not essential. In addition, a sound knowledge of ground-water hydrology and ground-water flow modeling are needed to effectively apply the methods presented.

References for cited material are given at the end of each major section. Good supplemental sources for the unreferenced material not peculiar to this text are presented as “Additional Reading” at the end of each reference list. It is expected that students who have difficulty with the material in this text will consult the more expanded developments in these supplemental sources.

Several people, in addition to the writers, contributed extensively to this text. Charles R. Faust wrote earlier sections on statistical review and basic regression and contributed several exercises, Steven P. Larson wrote an earlier version and documentation of the non-linear regression flow program of section four and contributed earlier versions of several exercises, James V. Tracy contributed to the documentation of the nonlinear-regression flow program, and Thomas Maddock III wrote the first version of the statistics review section. In addition, all of these people helped teach the training courses from which the present text evolved. Finally, the writers would like to thank the technical reviewers, Brent M. Troutman and Allan L. Gutjahr, for their many hours of review work and the secretaries, Anita Egelhoff, Evelyn R. Warren, and Patricia A. Griffith, for their patience and care in typing the manuscript.