Groundwater Modelling: An Introduction with Sample Programs in BASIC PDF
Preview Groundwater Modelling: An Introduction with Sample Programs in BASIC
GROUNDWATER MODELLING An Introduction with Sample Programs in BASIC WOLFGANG KINZELBACH Gesamthochschule Kassel-Universitat, Fachbereich 14, 3500 Kassel, F. R. G ELSEVl ER Amsterdam - Oxford - New York - Tokyo 1986 ELSEVIER SCIENCE PUBLISHERS B.V. Sara Burgerhartstraat 25 P.O. Box 2 1 1, 1000 AE Amsterdam, The Netherlands Distributors for the United States and Canada: ELSEVIER SCIENCE PUBLISHING COMPANY INC 655, Avenue of the Americas New York, NY 10010, U.S.A. First edition 1986 Second impression 1987 Third impression 1989 Libmy of Con- CarJoginngin-Publiation Lhta Kinzelbach, Wolfgang. Groundvater modelling. Bibliography: p. Includes index. 1. Water, Underground--Data processing. 2. Water. Underground--Mathematical models. 3. Water, Underground --Computer programs. 4. -1c (computer program language) I. Title. GB1001.72 .E45K56 1986 551.4 '9'0724 85-27526 ISBN 0-444-42562-9 (U.S.) ISBN 0-444-42582-9 (Vol. 25) ISBN 0-444-4 1669-2 (Series) 0 Elsevier Science Publishers B.V., 1986 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior written permission of the publisher, Elsevier Science Publishers B.V./ Physical Sciences & Engineering Division, P.O. Box 330, 1000 AH Amsterdam, The Netherlands. Special regulations for readers in the USA -This publication has been registered with the Copyright Clearance Center Inc. (CCC), Salem, Massachusetts. Information can be obtained from the CCC about conditions under which photocopies of parts of this publication may be made in the USA. All other copyright questions, including photocopying outside of the USA, should be referred to the publisher. No responsibility is assumed by the Publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any meth- ods, products, instructions or ideas contained in the material herein. Printed in The Netherlands VII PREFACE With growing concern about groundwater resources both with respect to quan- tity and quality, the need for calculational management tools is increasing. A number of excellent introductions to the concepts of groundwater flow and pol- lutant transport exists, a most comprehensive one being the volume 'Hydraulics of Groundwater' by Jacob Bear. However, for the student or the project engineer who wants to develop a model and do practical work on the computer, there is still a gap between the understanding of concepts and the ability to handle the actual computations. To bridge this gap, a course was devised and was held for the first time in the summer of 1983 at Qinghua-University, Peking, China. The financial support of the undertaking by the German Academic Exchange Service is gratefully acknowledged. The lecture notes presented here sum up the contents of the course together with some additional background information. The publication addresses itself to students and professionals in the fields of hydrogeology and civil and environ- mental engineering. Knowledge of the basic concepts of hydrogeology is assumed. Starting from that basis the book has the goal of enabling the reader to under- stand mathematical groundwa er models and write computer programs on his own. The way to arrive at that goal s to follow each step in the development of algorithms from the basic equations to the actual computer programs, work through the programs statement by statement, run the sample applications, create alternative data sets and continue further by mod fying the programs for one's own use. Nineteen pro- grams altogether are supplied with the text covering a wide range of numerical techniques commonly used in groundwater modelling. All programs are written in APPLESOFT BASIC and can be run directly on an Apple 11+ or compatible personal computer. With slight modifications most programs can be transferred to other microcomputers with BASIC capability and at least 48 K of central memory. In an appendix the necessary modifications for the use of the programs on an IBM-PC are indicated. Wherever possible, intuitive reasoning was given the preference over mathe- matical rigidity. I apologize to the mathematicians and advise the reader to dig deeper into the mathematical background as soon as his interest is aroused suf- ficiently. It is my hope that to some degree the volume will succeed in conveying to the reader the feeling that groundwater modelling can be fun. Stuttgart, November 1984 Wolfgang Kinzelbach 1 ChaDter 1 INTRODUCTION In many countries groundwater is one of the major drinking water resources. As such it must be managed and protected carefully if we want to put it to the most beneficial use. With growing development of the resource and with growing human impact on the aquifers, the management needs become more visible. Problems like overpumping of aquifers and pollution of groundwater occur with increasing fre- quency. To mitigate conflicts of interests and avoid severe, even irreversible environmental damage, we must be able to predict the reactions of aquifers to hu- man impact with respect to both groundwater quantity and quality. As regional- scale phenomena usually cannot be studied in laboratory-scale physical models, mathematical tools of analysis must be applied. Some typical examples are listed to illustrate the type of questions arising on a regional scale where mathematical prediction tools, i.e. mathematical models, are useful. - A water supply company wants to develop a groundwater basin in an alpine valley. Abstraction of groundwater leads to a lowering of the watertable which in course threatens to cause drying up of streams and detrimental effects on trees and plants in the originally wetland-type ecosystem of the valley. The licensing agency requires keeping the damage at a minimum. Questions arising are: Which maximum abstraction is possible under given constraints on minimum water-table elevations in the ecologically most valuable parts of the valley? How should the wells be distributed in the valley to keep the impact as small as possible? Is it reasonable to consider further measures in extremely dry years, such as infiltrating water from the river which flows through the Val ley? - A dam is to be built on a river. This will lead to rising groundwater levels in the surroundings of the dam. Questions arising are: What is the extent of flooding by rising groundwater tables? What extent of drainage measures is necessary to protect settlements and in which locations? - A well field for a water factory is to be installed. To guarantee good water quality a protection zone with no pollution-inducing human activity around the wells must be established such that water from outside the protection zone will need at least 50 days travel time to arrive at the wells, a time span long enough to protect against bacterjal pollution. A nearby river will infiltrate polluted water into the aquifer once the pumping starts. Questions arising are: What is the size of the protection zone? What will be the ratio of groundwater and river water at the wells as a function of the pumping rates? 2 - Due to leakage from a storage tank in a chemical firm, persistent chemicals have been released into the aquifer over a time span of several years. A plume- shaped concentration distribution has formed in the groundwater downstream of the pollution source. The pollution is discovered by chance in a well of a water factory a couple of kilometers downstream of the pollution centre. Questions arising are: How did the plume develop? What will its future extension be if no measures are taken? Can protective wells upstream of the water works help to diminish the problem? How long w i l l there be pollution at the water works if the source of pollution is completely eliminated now? Where should monitoring wells be put to obtain the largest amount of information possible under budget constraints? In the following chapters mathematical methods are presented which may be help- ful in answering questions of the type posed above. They comprise models of ground- water flow and pollution transport. The process of mathematical modelling involves a number of different steps. The essential ones are: - Posing the problem: This step determines the effort and accuracy needed. It also decides on the dimensionality and time dependence of a model. - Choice of variables: This step is the starting point of quantitative analysis. - Determination of the quantitative interdependence of variables: Equations are derived, usually by means of first principles such as balance equations or by empirical relationships. - Choice of solution algorithms and implementation on a computer. - Determination of model parameters: Every model will contain a number of unknown parameters wh-ich must be determined in a calibration procedure by comparing model results and field observations. Note that without field data there is no reasonable mathematical model. In the calibration procedure a sensitivity ana- lysis of the model parameters is useful. - Verification of the model: Calibration fits the parameters to an observed situ- ation. Therefore the model cannot predict the situation used for calibration. To check whether the model has predictive power, it must be tried on an indepen- dent set of data. - Application of the model. Modelling is an iterative process. Steps may have to be repeated. Even the starting point may have to be revised. Models are approximations of reality, not reality itself. Usually they are even very crude approximations. This should be kept in mind when using a model on which to base one's decision. Yet, a crude prediction may be better than none. Models of low absolute accuracy may still correctly reflect the different tendencies of alternative courses of action. Also, by introducing conservative assumptions into a model, worst-case scenarios can be used to find reasonable decisions even with inaccurate models. 3 The main merits of modelling are: Increased understanding of the interaction of simultaneous processes and influ- ences, concise problem formulation, focussing of interdisciplinary efforts into one goal, where the results of dif- ferent disciplines may be represented by submodels or even single parameters in a model, ease of comparison of tendencies, if not of absolute values, resulting from different courses of action, simulated in advance. 4 Chapter 2 REGIONAL GROUNDWATER FLOW MODELLING 2.1 SCOPE OF APPLICATION Flow models discussed in the following chapter describe regional flow. This means we are interested in aquifers or parts of aquifers with horizontal exten- sions much larger than their depth. Groundwater flow is divided into unsaturated and saturated flow. Unsaturated flow occurring in the unsaturated zone is essentially directed in the vertical direction. Major regional groundwater flow can only occur in the saturated zone. Discussion is therefore restricted to saturated flow. We consider porous aquifers only, as opposed to fractured-rock or karst aquifers. Yet, the methods discussed can to a certain extent also be applied to fractured-rock aquifers. Equations for three types of aquifer situations are given (Figure 2.1): - the confined aquifer, confined at top and bottom by impervious layers, - the phreatic aquifer, which has a free water table, and - the semi-confined or leaky aquifer, which is receiving water from or losing water to overlying or underlying aquifers through a slightly pervious top or bottom boundary. The observable variable which is described by the models is the piezometer head in the cases of the confined and leaky aquifer and the water table elevation, also defined as the water surface elevation in a piezometer , in the case of the phreatic aquifer. For both variables the term head (h) will be used. Flow is caused by gradients of the head. In the regional approach it is assumed that flow is essentially horizontal, this means ah/az = 0 (Dupuit-assumption). This is certainly not true in the vici- nity of imperfect wells, in regions with strongly varying aquifer thickness, in the vicinity of infiltrating surface water bodies, and in regions of strong ground- water recharge. These vertical disturbances, however, will usually become negli- gible over a horizontal distance of the order of magnitude of the aquifer thick- ness. We finally assume that the density of groundwater is constant throughout the aquifer (homogeneity of the fluid). 2.2 PHENOMENA TO BE CONSIDERED AND BASIC EQUATIONS OF FLOW The flow equations for all aquifer types are obtained from two basic principles: - continuity and - Darcy's law. While continuity demands the conservation of water mass, Darcy's law states 5 a) confined aquw piezomder head h groundlevel impervious top cquifer impervious battom 2 1 - groundlevel watertable elevation h unsaturated zone groundlevel overlying phreatic aquifer / I semipervious layer h main aquifer ~/////////////////////////////////l/m/F vious bottom Fig. 2.1: Aquifer types discussed 6 that in an isotropic porous medium the specific flow rate (filter velocity) is proportional to the negative head gradient. In horizontally two-dimensional ground- water flow this is written as with 7 = (v x1 vy) and ? = (a/ax, slay). The proportionality constant kf is called permeability. In anisotropic aquifers a generalized form of the Darcy law can be applied. v = - IK .%V h IK is the second rank tensor of permeability. It allows one to take into account the fact that in an anisotropic medium the direction of flow may be different from the direction of the head gradient. The differential equation of flow is derived by taking a water balance around an infinitesimal control volume, which extends in vertical direction from the top to the bottom of the aquifer. Consider the confined aquifer first (Figure 2.2). Over the time interval [t. ttAt] the net flow entering the control volume must balance out the increase in water stored in the control volume. Flows entering the control volume are counted positive, flows leaving the control volume are coun- ted negative. Flows to be considered are the horizontal flows and the recharges and abstractions made through the top of the control volume. r SAX& (h(t+At) - h(t)) m is the thickness of saturated flow. S is the storage coefficient; it expresses how much volume of water can be stored additionally by compressibility in a column of the aquifer with unit cross-sectional area and height m if the head is increa- sed by one unit. q is the recharge/discharge rate per unit horizontal area. Dividing equation (2.3) by AtAx& and taking the limits AX -> 0,Ay -> 0.At -> 0 we obtain the partial differential equation: ah aV.( m ;; ) + q = s-a t (2.4) Inserting Darcy's law yields the flow equation expressed in the variable h. Flg. 2.2: Control volume for the derivation of the horizontally two-dlmensional flow equatlon h (t+At) watertable at //’ time t+At h (t) watertable at time t water volume stored I I bottom of aquifer -dimensional model of the Fig. 2.3: Storage mechanlsm in the horlzontally two- phreatlc aquifer
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