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' •• • '·. ••• • • • ' .¥ > ' - .. ,. .. ·' ' . . . I . LIBRARY --;- TEXAS. WATER DEVElOPMENT BVARll ' AUS liN, TEXAS I TEXAS WATER COMMISSION . Joe D. Carter, ~Chairman 0. F. Dent, C~issioner · H. A. Beckwith, Commissioner
. ·~:· :.. ; : ; .· _...:._ __ _:_ __._, , Report LD-0165
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.. ~- •.•' .. ·.::. . MANUAL Of ,, COMPUTING AND MODELING TECHNIQUES . . . I . . ' . . ·. ' AND THEIR . . .. ', :~ ' .. . ~·r -(· I ' \• '•' '' ·:. ·: ··:-_ i'.;'' '.,APPLICATION TO HYDROLOGic· STUDIES.:. 1·'" ',' · -:: -: · .. i;. ~ • ' i · .. .-. . c , r. <'
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By
J; Russell Mount :. . -· ..
·,
·' --,__ __ ------
., •.- January· 965 . PREFJICE
This manual is written for employees! of the Texas Water Commission .eng_a~ed.
in problems 'of water resources, with the hope that by gaining a basic under-·· I . standing of computers and models they rna~ develop new ideas which will lead to
increased efficiency and output of worlt:···-Much of· the manual is devoted to ex-·
plaining the basic principles of digital [computers and ~nalog computers ·and
models mainly for the purpose of removin~ some of the mystery surrounding these. ·
.devices as they generally'appear to the 1lyman. Other parts are concerned with
I hydrologic applications of advanced matheratics. ··The material presented is
· d;irected toward problems in both ground-water and surface -water resources; ·...... I heavier emphasis .is placed On grqunci-wateT applications owing to the w-.ci ter- 1 8 . I . . . experience, howeveT almost al.i material. p~esented.I is from. available.. literatu:Ce...... •. Because -che aim of the manual is to present information in an understandable·~;,
way, complete development is r,ot pu:csued:! An extensive bibliography at. the..
end of' the manual may be useful ·co the relder who wishes to investi~a-ce -nTI3
area of interest more thoroughly. This.+esentationis: nota ,technical paper:
but rather a. comprehensive teaching aid,. (hus in order to· fa,Cilitate clarity. references are not· do,cumented. I ,_ ·. I
\ ' ..,. , . ·· ... J. RUSSELL MOUNT ., , . ' ' .. .,-., .• ,,,:.
.. ·• \ \ ;' .
S/0. 78 M86m .Cof· .(; . CON'f'ENTS
Chapter ··Page
I. INTRODUCTION .. 1
·· ----~ -II.·· ··· THE HIGH. SPEED ELECTRONIC DIGI L COMPU'l'ER 5 t. , ,: . III. '· · ·ANALOG MODELS .·· .. 11
IV. PARTIAL DIFFERENTIAL EQUATIONS .14: ·c v. FORMULATION OF HYDROLOGIC PROB INTO ELECTRIC- . ANALOG MODELS 24
VI. ANALOG COMPU'l'ERS ...
-,~ ',
VII'.· .EXAMPLES OF S.OLUTIONS OF HYDRO GIC PROBLEMS .WITH • c"•·. THE USE OF DIGITAL AND ANALO TECHNIQUES ~ ". .,,·!
Digital Computer Methods . • ,·' · •. 37 ...... -1 • ._, ~· 40 .·· ,.. Analog Model Methods ...... · . . .' .. .- ·.·'
·, i· ' -~ _; ·.. Analog Computer Methods . .A3,,: '•. "'' ':••· CONCLUSION • • • . . ,... ' 48 -.. ,.,., ...... BIBLIOGRAPHY ; . ; • • . . . ·-· .. ,, ...... _. I ,, ., __ ._ '" .. ,; .. ;., ...... ~1 ., : ,~.
,. . ~.-.' ,.. .. ~ . ,. . :.. ,, ' . ,~- : .-. ,, ' . :' .,_·- (<_
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ChaptJ I
INTRODUC 'ION
Since World War II considerable prog ess has been made in the development . . ' I of automatic. computers. and their applicatJ,ons to engineering processes, owing
to advances in electronics technology. T e computer field rapidly divided into
two branches, digital and analog. As eac computer method gained adherents . ' ' . ··•.·· "'considerable r'ivalry develoJ?ed. However, this seems to be only a temporary ... . . ·.situation; and-as engineers and scientists become familiar with both methods;
·'the~ generally-realize that :a~h mecho
,. W~ {irst le'a~iJ'~d to coun-c by us1ng fir,gersj, and probably' the decimal number· , . P. ~·-~sysi~~ "is an,.outgTowth of primitive man 1 s lse of his fingers in coun·cing. The Chi'nese ·~'tacus is another primitive form o~ digital computer. More familiar ~-:··-to us are desk ad!li.ng machines and calcula~ors, complex d.igi tal· computers .. ·' c However, the term digital computer general~y implies a more sophisticated '
machine called "electronic (brain", the amabing device which threatens to replace .. I so many clerical jobs and create unemploym~nt problems. ;r:ts. desirable features· . I are tremendous speed of computation and flbxibility. ·The .important thing to · remember _about a digital com~ut~r (~~d thib. applies not only· to. the _simpler ones· .buc also to the advanced high-speed electr~nic digital computers) 1s.thatit.· . ' . ' . '·· ':·'
·.-."' deals 'in digits and provides rather exact numerical answers. Again, a digital
computer counts. Its accuracy depend_s up, n the number of digits provided for ... - - "• --.--- processing numerical data.
_Next consider an analog computer or an analog model. The word analog , I implies similarity, or analog. Two systen\s are said to be annlogous if there is I a one-to-one correspondence in their physical properties. I Analog computers provide answers to 9alculations by dealing in a physical·
. systemanalogous to the system which is bJing studied. It was pointed out· in . I the previous paragraph that digital compuJers count. Correspondingly, ANALOG , . . . I . COMPUTERS MEASURE. An example of an analqg computer is .the gasoline indicator
.in an autombile. A float in the gasoline tank is mechanically connected to a
·:,.' , device which controls electricity flowing Iinto an electrii: meter on the dash
·. board. Actually electric current flow is being measured, not gasoli~e. But
the rate of flow of 'electrical current is !analogous to the amount of gasoline I in the tank. Another example of an analo~ computer is the slide-rule on which '
mathematical operat·i:ons are performed by 1easuring lengths along ;;he memberos of.
, , , \' the rule. The term analog computer, howe.Jer, generally refers to complicated \. :. \devices called. electronic differential ana!lyzers; they perfo~m mathema'tical' . I . operations by using laws of electricity. Their most common applications are in
··solutions of differential eq-..iations.
An analog-~ differs from an analo computer mainly in that the computer·
is\ a general-purpose devi_-ce .. _designed ··to so[Lve mathematical :relations, whe:Ceas a . - I . . . -·._ · · model is a special-purpose device created to simulate e. physical systoem af_inter-'
··:; est; hence an analog model is sometimes called a simulator. ·, \ i I . An analog computer uses physical laws! for mathematical purposes, but an I . ; ANALOG MODEL HAS A DIRECT CORRESPONDENCE ITS PHYSICAL PROPERTIES TO PHYSICAL
PROPERTIES OF ITS PROTOTYPE •.
! An analog'computer has·a bJ:'Oad range f :utility in mathematical :problems'· . . . ~. . ·····-.:' ~ ._.: ',' >- .. . . •. 2 . ·related to engineering,.physics; chemistry and so on, but an analoc model is
constructed only to provide answers to a specific problem.·
A scale model·may be considered a spe ial ty-pe of analog·model. Vlell-
.' publicized scale models are the river mode s at Vicksburg, Mississippi, where·
specialists of the Corps of Engineers atte pt to predict, among other thiriss '·
·a r:lver's hydraulic response to proposed m·difications in its channel. Aircraft ' designers use scale models exten.si vely in ~ind. tunnel studies. In scale models. we observe that the p~ysical system in prototype .and mode1·
are the same. The only changes are in thelscaling of dimensions. But with
, analog models other than ·scale models the hysical ·systems are different. For
example, flow of heat may be: simulated or ode led by flow of electricity.
·,;i A summary of differences in analog and digital processes follow .. Digital
· 'techniques. Digital computers are more ac urate than analog computers and models, and where accuracy is crit:l,cal, d~~~~al·c~Jputers· are recommended. . .. ,. Computational processes in a digital Jomput~r differ .from those u.sed in: . . I ~v '• . ·manual methods and 'the. scientist or engineer is separated ·from his p;:-oblem . I H . ;~::. rduril}.g the :computational phas'es; but with ~-n analog computer or model _the_ -metp?~s !'-'of computation aTe laws of physics, and. ar, therefore quite direct. In this ··::; · sense, an _analog computer or an analog mod~l. m~y be distinctly advantageous· in,-.·. . I ·that it Can provide a fJet~er insight into the beh&..vior of .a problem than a I ·digital computer. I In the past few years no'table advances have been made in digital computer· i technology involving greater computation sr1eed, larger memory. storage, simpler programming techniques and more efficient Jircuit d~sign ..:_Comparable. advances I1 . . have not ·been made· in analog computation. ~Therefore many problems which .previ- . . I ously were considered solvable only by ana!og methods are now being solved .b:;; . I . I . . digital methods ... In particular there is a tendency. toward preference of digital 9- computers over analog-models owing to tj greater accuracy obtainable with ··. • . . I digital equipment; A singular disadvant ge in using analog models in the large ·investment' in equipment which has applic tion only to a· specific problem. ·.·· .,· ... .-·. · . 'i '. ·.' >, ., '• !.' '·· . •. '• \ ' . ·- :.· '.· ·' J~ -, .. ,,- " ' \ ....._, ; ~ ~: .,. ,..:_ ,'.·' ' ~ .. . . : . '·"' ':;.. -: ;· '··. .. :.. --· ·, (i·- ··t .. :· -.: ..\ "· "· ';.. I ' • \. ~- ,· " ., . ·'-.• .·· . ';·:- ,,: ···- . '·. .... .;. 4 '. ~· ' . ,.· ' . ·.·. - ·.. ·'•' •. ~· ' . I<,• Chapte . II i THE HIGH-SPEED ELECTRONIC DIGITAL COMPUTER I The high-speed electronic digital c~mputer is often called simply a digital computer. It can perform a tremendous mkber of computations with incredible I _speed, which accounts for its acceptance lin business and industry. However, because of the machine's structural complexity many are apprehensive toward .developing an understanding of it. It iJ hoped that t~is chapter will overcome· i s.ome of this fear by explaining generally what a digital computer is, how it' I works, and what must be done to .use it. I - _<$- First consider briefly the method 04 operation of a digital computer by comparing it with a desk calculator. Su~pose we have numbers to be used in a I' computational process. Now, in using a calculator, we must dec-ide which number: i ·"'. to enter first, which second, and so on . We must also decide whac operations .. _\(addition, subtraction, multiplication, Jnd division) are to be used and their sequence. Computation op~~ations. are th~u performed in the desire~ sequence. · . I -- The answers o·otained must then be properly recorded, manually. I If we wish to use a digital compute~ for the same purpose, v1e atill d.ecide ' 1 in advance which operatious to use and id what sequence; furtheTmore we Qecide on the form in which the answers are .to -J,e presented. liowever, all these de- ' "·cisions are converted into instructions, ~called a program, which the computer . reads, memorizes;. . - .and- executes.. . The. computer-I. is also. given. .. . the .numbers or data to· be used. The; i;>structions and data arle generally presented on punched cards or magnetic" tape.· · The principal distinctlion. between the calculator and the digital computer methods. is in,the speed ~f comp_uta~io~:. ·. Witli the ;digital: : ' \ ...... computer, computation is extremely. rapid although considerable time is required to devis'e a program. The decision to us either a calculator or a computer.. · should therefore be concerned with the t tality of time and. costs involved ..A ·.computer would be preferred for instance in problems of a recurring nature. A. distinct advantage in using a computer i that by reducing the time required .. for tedious manual computations, morale ,f personnel is considerably improved, 'and engineers and scientists can then deite more time to technical duties. A digital computer can add algebraically, make decisions, and follow a logical set of instructions. The method ~n which a computer' solves a problem is to read and store instructions, read a~d process data according to the in: ·~ ·structions, and present the results. in thtl . desired form. ,• . . Having considered the computer's metod of operation we now investigate. ~,~. '' its mechanism. The .mechanism of a desk c lculator will be compared with the. mechanism of a digital computer. I . -·-·--- .....• I i· Anyone familiar with the desk calcu1¥tcir ·realizes that numbers appearing l . in the register windows of the calculator are actually digits printe vidual cylinders. Each cylinde;? contains 10 digits, fTOID 0 to 9 (this is £. basically the decimal number system). These .:!ylindeTs are turned_ by mechanical_-. I " '':, pulses generated in the machine's computimg mechanism. A high-speed digital . . I . I .. , COTI).puter use? electTonic pu~~es rather th1n mec~anica:r' pulses, and. in the more ' t. · . advanced computers millions of pulses are Igenerated each second.. These pulses . ;·.: ···magnetize tiny iron cores and open .and cl~se tiny electronic switches connected -~. in intricate combinations" Some of these I tiny compon~nts are· used to stoTe· · ,, information (the memory), others are used. lin computing processes. I - . j A digital computer does not use the decimal number system; it uses tFLE! ~ binary number system, '1hich contains only 2 digits. The reason for this is that [{-' ,,' L'· in electricity andmagnetism there are onl!y-2 stable states .. Eiectricity or,.· ,,' 1' magnetism is eitherposJ,tive or negative. An electric switch· is· either on· or· off'. ' v· ,,~}. ;~ \; " :\ A comparison of some decimal numbers and corresponding binary numbers are shown I below: Decimal System Binary System· 0 0 l' 1 ·2 10 3 .11 4 100 ..-·· 5 101 ; ·. 6 110 7 111 8 1000 9 1001 10 1010 11 lOll In manual computations the binary s~stem is much more c~~bersome chan.the •' · decimal system, and it would seem impracJical to use such a system. It pre·sent~ • ·,no problem, however, to an electronic di1i tal computer, because of the speed in 3 which the binary system can be nandled. !Digital computers are designed with' I . • ·. ·. principles of boolean algebra,. an abstradt al'gebra, which deals with two sets. Although hund.reds of thousand.s of eJectrical parts are used 'in an .ad~anced digital computer, they can be packed intJ a relatively small space, owing· to . recent advances in miniaturization tec~nirues in electronics·. .. . One. would· suppose that in order: to ble able to use a digital computer he .would· have to be· acquainted .with binary a~itbmetic and boolean algebra, ·However;. . . . . I ...... tbe computer design engineers have assisted us by placing between the operator· and. the compute_r· a ~~b~ack.;:boX.n ., ~ca·lled a comJ?iler, ~whi.ch trans~at_es d..ata .from .;.·-":- decimal to binary ·system and codes our "h\nnan'' instructions into machine . language, Because information must_ -~~ .~+~~en~-~0. to the compiler in a standard form, there have been evolved many compiler languages. pome compiler languages I require ·a great deal' of knowledge about tre computer's system, but there has . been recently developed for scientific anr engineering work a rather straight- . I .. forward lil.nguage, 'called FORTRAN (FORmula TRANslation). One of the principal LIBRARY TEXAS WATER rmnMI<:<:inN · advantages of FORTRAN .is ·that besides beJng easily learned it has become so· standard. that now almost all c<'>mpU:ters fl scientific application have FORTRAN compilers. One disadvantage of FORTRAN is that it cannot utilize all of a computer's capability. By being· removed from the a solute computer language, and being dependent on a compiler language, some o the computer's resources are sacri~ ·. ·ficed. ·A FORTRAN compiler is to a digit l computer what an automatic trans~ / - "'·' ; mission' ·fs to an automobile.. The ic transmission is easily learned ·and ·.·is entirely: satisfactory for most purposes, yet no thinking driver would . '/.. : ~;; ·: .. ;•, ··· :e~t~~ the Pike's Peak Race with a car so equipped. There are some tasks· f_or . Similarly, although many problems can be solved on a digital computer by using FORTRAN there are also some problems which can· not be solved unless the absolute . I . machine language or a compiler. language •ealing more intimately_ with the machine · . than FORTRAN is used . . . ·{ In general, the difficulty of creat'ng a. progrrun'depends on the complexi~y ., : of ·the problem. Many problems I · services of a profess~onal programmer ar, ~warranted." However 'if the "'esults.; j, '·:. ·are important enough; t~e cost of the sezjvices are justified. . . I '·•t; ,; Most computers have built into them !special subrout:ines.j Whicl1· are ·.upre.; · }~ ·'··· .-.·~ • fabricated" instructions for commonly us,d mathematicai operations such as .. : ~ .. square. root_s, trigonometric functions, anti ~og~rithms .. A simple _code· .. in< a . I I progrrun.calls the.desired. subroutine. intolI . action. An elementary example of solving a ~roblem by the· use of a digital comput_er I . . . using FORTRAN.' language is. shown below,' .The program .can 'be used as often· as . ,o; desired- it is not specific. If the s~e operation with different data is to· be performed later, only the data needs tb be .~hanged.~· The pr~gram ins~r~c'ts :. the. computer· to· ~alculate the sum of two Lho~e ·numbers which ·have' ~o ,more than,.· ,,.· .. . ·';'······ - J '• ·----:--:----'-'-·---··. -·--· -'-·· ·- - 8 ·' 1 i p~:~ c-, ·, . I . . I I I .J 0 0 0 0 0 0 0 0 0 0 0 010 Ol 0 0 0 0 0 0 0 0 0 DD DD D0 D0 D D0 0 0 0 DD DD D0 DO 0 D D0 DD 0 0 0 D DD DD 0 D0 D0 0 D D0 D 0 0 0 DO 0 DO 1 1 l t J I 1 I I 111\1111 !liS 111111111021 tl 23 )4 :IS l'fllztl'lllllll! :P :H-" :113131 Jlq4J414l"'~· 1141 4t ~SI Snl5oi511!JtSI !oiSIR"Itll ... I1WIJJSM 1011 IZIJ N II II 11" n R 1 I 11111111 11 I 111 I 1 I 11111 I 11 11 I I 111 I 1 I 111 I I 11111111i 111 1 II I I II II 111 111 1 1111 I 1 I I 11 1 I ·, 222222222111111111111111121111111112111111111112111111111111111111.11111111111111 ~ 3333333333lllllllllllllll333~lll33333333333333333~33333333333333333333)!!3!!!!!3 44 4 4 4 4 4 4 414 4 4 4 4 4 4 4 4 4 4 4 4 414 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 414 4 4 4 414 4 4 4 4 4 4 4 44 4 4 4 4 4 4 4 4 4 4 4 4 14. 55555551555555555555555555555555555555555555555559555555555555555555555555555555. . I . . . . 6! 6 66 6 6 6 6 6 6 6. 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 66 6 6I 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6.6 6 6 6 6 6 6 6 6 6 6 6 J 1llll J J J llll lJ 1 J 1'1 Jl 1 J J J J J 1 J lll 1111117 1 J J JJ J J 77117 J J J 1 J 1 J 111 7 111 7 1 1 111 1 1 1 1 1 1 1 ·1 1 . . I . .. 18188BIIBIBBI88I888888!8888888888BBIB818!8888!81B!B8888BBI8888BB8888888881888811 9~ 9 999199 9 9 9 9 9 9 99 9 9 9 9 9 9 9 9 9 99 9 9 99 9 99 9 9 9 9 9 9 9 9 9 9 9 99 9. 9 9 9 9 9 9 9 9 99 9 9 9 9 9 9 9.9 9 9 9 9 9 9 9 9 9 99 9• 1214lllii"~112»Mnnnnaa~nuMnanaa•nuuM~sn~,.~~uu~e•uaa•~nn5oiM!.I~MHM~~bMe•pannJ1nn~~Nnnna ....Of'l 'i.: Punched Card for FORl:RAN Statement No. 1 in Example Program ',,I ) ; ..! ,,.•• J K I I .. ' .·.' ·.·!·,,..··•·· ·' . · OOOOODIOOOODOOOOOOOOOOOOOODOOOOOOOOOOOOOOODOOOOOOOOOOOOOOOOODOOOOOOOOODOOODUDOOO f. 11li'IIJII11111lU141!nnnn~~~nNna~al'flll~nn~""M~:.•~••Quauauaa~~"~5oiMM~MM~ItGaMn•vannnnnunnnNnu ·1l11111111111111111111111111111111111111ili1111i1l111l1111f11i111111111ili111ili ·. r ": 21111121212111111111212 2 2121111212 2121212 2 2 2 2 2 211~ 2122112 2112 2111112 2 2 212 2112 2 21 ••. r '· •. ; 3 3 3 3 3! 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3) 3 333 3 3 3 3 3 3 3 3 3 3 3 333 3 3 3 3 3 3 r. 33) 3 333 3 3 3333 3 3 3 3333 3 33333 J 3 33 44444444441444441444444144444444444444444<4444444~4441444444444444444444414<<•11 .. s 5 5 5 5 5 5 s 5. 5 5 5 5 5 5 s 5 5 5 5 s s 5 5 5 5· s 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 s 515 5 5 s s 5 5 5 5 s 5 5 5 5 5 5 5 s 5 5 s 5 5 5 s s s 5 s 5 5 · . ·· . &&js&ssssussssssss66&&s&66ssss6ssss6ssss6ssssss6s~sssssss&6sssss6sssssssssssssss I .~:. llllllllllli1Jllll.1777117JJJJJ171JJJJII771lllll17~lllllllllllllll1Jilll1JI1lll1j ·l.· , , 8818 818118181818 8 8 8 8 8 8 8 8 8 8 8 8 8 8 818 88! 8 811 !i 8 DB 18 81~ 8 8 8 818 818 8! 18188 8 8 8 8 818 B8118 8i ~ . ·aa9aaUs99asas9999999sss9sass9999asssss9asass99999~ssssss9ass999999sasassssssBsss . . , ...-. _:.. _II_'-~ I I!.! :_o'! 111Z!SI411111711·11·,n~!I:K;r,az:rD21JIJIUJIMB:.U_JI •• 41414t .. Ga4J_. ...ISDI1~5:1~AASlA5t~l1121lM_._..J.An111T1llnMfll~n~tiiiC' .'-·: ~ 1 11 -, .. ·.," ~ . '· .,.', . ·' t' . I . . ,; ~· . ··~ Da~~ •. car~.}or ~.~l'ers J and _K,in E> ·. punched card for statement 1 is .illustrat d on the facing page.) . ··•···. ,:,: .1· -'READ 2, J, K . .... ' · .· 2 :.FORMAT ( 2I~) :·-- . .. 3 :,,1 = J + K ./Ji .PRINT 5 , L ·· . ,, ·5 .. :.:FORMAT ( 4X, I5) 6GOTO.l 7 END Statement l instructs the computer to read from a data card two numbers, which it will call J and K, according to ~specifications of statement 2: Statement 2, a format statement, sperifi~s that J is found punched_ within the first four columns of a data card (the first column· is for a plus or minus · sign), ·and that K will be found punched ~thin the second four columns. ·The· letter I is a code which signifies that the numbers are integers " decimals are· not- used. (The punched card for data in bhis format is shown on the: facing ..page.) Statement 3 instructs the computer tr calculate the sum or' J and -K and to call the result L. · Statement 4' instructs the computer to print the number called L, in ac-. I . . ~. cordance wi ththe format specifications o~ statement 5 ... Statement 5· specifies that the sum (L) will be printed in· a block 5. spaces wide, located 4 spaces to the right of thl left hand margin. Statement 6 direccs the computer to return to 'statement l and read another data card,_and h7nce repeat _the addition and print-out process. --~ .. ------.Statement 7 ·is necessary for the prober reading and compiling of in- . ·struchons. It is a signal to. the computbr that there are no more instructions ··_and te>. _proceed with the. executio'n of the broblem. . ~.. .• ...... I . Digital· computers .ar_e quite expensive and are generally purcha~ed only ·by large eo.rporations. Smaller !:!Orporations either use computer consulting servic• or obtain computers on a rental basis.· C mputer-time ·rental· rates_ range f~om .. - 9 - about $35/hr .' for the smaller computers slch as Control Data Corporation· Model 160A to about $550/hr. for che largest co~put~rs such as the International Business Machines Model 7090. . '. " .[.', ·.· ', ·· ... · -.' ' ,._. .\' ''\' ·_( '• ~ -~ .. ,_ ·' :.. :,·. --~,' 1. '•· "•, ·. :: .• :! ·., i...... :··. ;.:, . .' ~ .. _, ,. -?'"". :') ::· .. . !-:· . \ ., l •',' '' ._ •.. , '·· '?' • '. •·. :-- '~ .. .,...... ~ <,;._ ;- .': -~t·,; __ •'i I ...•-- \,. ., '··· - lO Chaptef III ANALOG Jl ODELS Before proceeding with the discussi n on analog models it will be helpful I ~--- to review differences in the digital and lanal~g systems. It was pointecr-out -- .. that the principal difference between digital and analog processes was that \\igital types use counting techniques,_ wjereas ·analog types use measurel!lents of physical quantities. Furthermore the ac uracy obtainable with analog equipment. 1 is-limited by the quality of its compone ts. There are, in general, two kinds of analog systems: anal9g models (simu- lai;ors) ·and analog computers (differenti 1 analyzers).· In an analog model:; a ·_prototype physical system, such as a gro d-water aquifer, is represented or ·. modelled by another physical system with similar properties, such as an electric circuit. The model is subjected to cond'tions analogous to those encountered. in ,• the_ prototype,_ and the behaVior or react'on of the model is studied and/or measured. An analog computer has_features sim·l~r to an analog model. However an ,.analog model is built only for a specific problerr,, but an analog computer. is a • general purpose -t-ool :designed. for a wide ariety of. applications. . Analog come · puters ar.e )lSed extensi vely.in: engineeri~g problems. to solve.. ordinary differ" '·ential ~~uations and- are discussed more sbecifically ·in Chapter VL. l.. ~ I - -· ,:,_ · ;- ~r:alcig- models are! used: in' many bran+es of engineer.ing reseai-ch; · The. choice ;:;~nd d.~ sign ~f the mo - 11 -. I I properties; only the dimensions differ. IWhm the physical systems diff!"r, electric- or mechanical-analog models are generally used; An example of a scale model is a sandi tank, in which a glass box is filled with sand, and provision made for water Jo flow continuously through the s:~d. \ Various conditions may be imposed, for iJstance water may be pumped from a simu- 1 . lated well, .and changes in configuration lor the water levels in the sand ob- served. Often this type of model is use~ only to illustrate the tre·'ml:vior of the prototype system; measurements can blI misleading. I \ An example of a mechanical-analog modelJ is a membrane. To illustrate an - .. . I . application, consider a well pumping fro~ a water-table aquifer. Water move,._.-- ' through sand toward the well causing the [water table to form an inverted cone· \ 1 _,\with.its apex at the well. The same shaPe occu_rs itJ. a membrane ·when one presses a s h arp 1ns· t rumen t aga1nsv· + 1·t . ~=1 as t'lC properI t·.1es ot" ·the mem b rane are ana 1 ogous . . to water-transmitting properties of the lquifer, and the magnitude of the.poi~t .. force applied to the membrane is analogous to the rate of pumping from .the , aquifer. Electric-analog models, however, ap ear to have greater utility than 'mechanical-analog models because electric properties have analogs in so many ... ;-·.~other physi,cal systems .. FUrthermore)' ele tric components are ~elatively. in '·'.' expensi ~e, their properoies are not affecfed. by gravity, substitutions or. ~-.:_ modifications are easily accomplished., and an· electric system can. -be construc~eci· for convenieno access in measuring. Specffic discussion of analog models in_, . this manual is· confined to electric. types I. · ·. Construction of an electric-analog mpdel req_uir~s that one be familiar not· .: only with the physical behavior.. of the prftotype system but also with electronic. ·.~· t~eory ~~d techriique in order. that parts ~or the model. can be properly selecte.d •and assembled; , ~he gen~ral p;ocedure is lo. construct the model ·(the; sy~~em), observe or: measure 'the response' '.:. . ,·· - 12 - ' " I Electric-analog models are used in two types of engineering problems, synthesis problems and analysis problems. I A synthesis problem is one in which the excitation and response are specifiedjand it is desired to determine the . ~- system conforming to these specifications.! An analysis problem, on the other I hand, is one in which the. system and exci11 ation are specified and it is desired to determine the response. I . In a synthesis problem the usual procedure is first to constract--a model' \ by estimation of properties and to test tie model's response. Then the model i·s ·repeatedly modified and tested until i1s response conforms with sufficient ·accuracy to 'the desired specifications. I · Electric-analog models provide solutJons to partial differential equations,· \\a subject which will.beii.~~eloped in. the Iext chapter;·· . ~- "'· ._ .. ·' J; ·,, ' \ ·: l,. .. . . -; ,. J •'•, ·-··. . . .. \ ' '- ~ ., 0"' ····:1 ''• ,-,_ ' ·:· "'(. .·-- :r. : ," . ' -~ ' ., . ··" ;. . . \ .. · .. ;., .·:. ... .-.. ,._, t-• •. _ -., '-· ·';· ~-- . . - j -~.- ...... ::~ "\'' ; : ~· ...... • \i .. -t'· . , .. ,. ,·, ' ' :.·"· ·' l·, 'r,' :-' .• ,, . '' ' i.. '• ~ ,. . ,.. _,_.•' ..•· _,' ~ ' .., i ~ •• ·\ .. ' .., .-, .. ;, : "· ·-:: '.• 13 - .-., Chapte:r:i IV PARTIAL DIFFEREN IAL EQUATIONS Partial differential equations are ommonly used in.physical sciences and engineering; mathematical methods ling them have principally been an· · . outgrowth of scientific research. It is the purpose of this .chapter to attempt: . only a physical explanation of partial a· fferential equations; mathematical precision is relaxed.· Moreover, in many cases ·partial differential equations cannot be solved by classic mathematical techniques, but only by approximation_ ·:·--methods.· . Nevertheless, the mathematical concepts are useful for gaining a .;..: : .. USf'd types of partial differentoial equat;ions are given in this section. .. · Before introducing par.tial differen+al equations it may be. helpful-to· point out some elements of their broader mathematical field, calculus. First observe that an· algebraic eg_uabion in two. var~ab~"~ ( fo~, ~~ample; . ~ :. ., _and y) can ·ne displayed in graphic form. Some equations g~aph'.as lines, :other·· "' ...... :.·. . . ': ··.. .: .' '". ' .. ' : .. ·· .. ·, ·. · ~·a_~··~~v~,~:-~~-}l;;·F9,;. ,,t~:~i -e~uat~'?n A = y gr~~~s·~:~·s :a::~~~~ 9 ·-, .. ··: ··:.··: .. .' . ,..-' \ ' ' . ···.·''' ··•', ... .-:: I . · ','' ···-. t·. ,, '.· ',:-~\~ ' •.· -.. : .. . . •{; ' ,... ~' . /';~ ., :·· "-· .,:·· -·.i _;·. -,. ,, .. · : ·. •' . ' ~ '. -. . ' .. '· .··: ~ 14 - . \ and the equation .Z. + :1 = 1 graphs as a circle. y \ . In. calculus,. where equations· graph as curves,. the· cilrves are considered . to be compose line segments tend to zero. ,;,'' .,. However, .in many,engineering problem[ only a graphical curve form is knC!wn; 'the associated equation .is not available. In such problems, satisfactory .·· :{·' ,, ' solutions to p~obl~ms may be· ob'Cained by pproxima'ting the' curve by line :.-segments of fini'te length, (The smaller ttl e line segments, the· more acc~ate :. · · . .'·the·approxima'tiop.). Such methocis of solut on .. are: calle '' .· example of'. a f'ini te. di f'erence method is to. calculate the·:'';:. '· a;proxiJilll.tigg its;bo daries.:as straight. line ,seglllen'cs~ •. ' . . '"···· : ;: . ·-~~ ,> y . ~ ._: ·.. : y ·-~:: ~-- .;; ~~ . . ,: . ' ;• '. '•' ' 4 .... :, .• ·/ - 15 - The circle on the left is approxima t,ed by the figure on the right whose area can be calculated by ."coun~ing squ~jes". . I Finite difference methods ~an also l:lle used to approximate algebraic ... equations in three variables, which grapH as surfaces in space. . . . . Differential equations is a disc.iplilne which deals with differential calculus. Differential calculus deals wirh expressions called derivatives. A dcrivntive of an algebraic equation describes the inclination or slope of any I tiny line segment in the graph of the cur:ve. Solving a differential equation consists·of finding an algebraic equation whose derivatives satisfy the differential equation. Partial differential equations invol e algebraic equations in more than,_ two variables .. · ,They are composed of part!Lal derivatives, which are derivativ:es . in:rc>lving only two of the variables .. Pa+ial differential equatio~s are en-·:'. coiinter~d frequently in proolems involving space and/or time. . . . I We begin with ·Laplace's partial diffbrential equation, ··--. Expressed simply, Laplace., s equation mean-s 'that· the Tate of inflow equals I the rate of outflow. If we deal with a hydraulic system -in which the rate of movement of water flowing into the system\ is the same as the rate of movement . of water flowing out of tne system, then we say· Laplace's equation is satisfied. Furthermore, 'ohere is no cnange in the ambunt of water stored in the systoem. · To· I ~ite-a common example, if you spend all ybur montnly income and your savings·- 1 . . . . remain intact, then.yourfinancial systemlmore .or le~~ o~eys Laplace's equati~.n._ In order to. see ho:W·Laplace's equation applies,· consider·a water-level . "' ; ' ' .,. .. ,· contour map: • . , ''· · .: . ,, './. : .. - 16 - I Flow -"--- w E I I I I s I . The flow is clearly in .one principal direction, east. Water enters the. west side of the small outlined ~ox~ andjexits from the east side. The box ·has width; W, and length, L, as lndJ.cate . .· . ' ' Next, change'the·map directions E ad N to graph directions;x andy.· . ',,, y Flow _,_. n--r-iW'' I ·. j ' I ''• ( . . ., .·.·_ -:-'-+-:-'--1-:-1-'~=-,_-,_c_-"'J'-++-'--'----- :< ·..... •,";·" '.,' ·- -:;- -: .'.': I '"", '\." . , -~ .·; V- ,. .;.: .. •·.··.-.r'. ccinsider a cross.' section. from eft to right. across the 'small boX, ~ow. ,· ~- ,. ... ' ' ,·.i. :,_,. \/~ }~ ' ~- --~-\ . .,._ ,... ._. ( ,<·, i.:ond Surface·. ---r-f!-1 _·_c_warer Tablej . ' . '.; . ~·:~ o·2 ., .. ·-·t ·'· .. ;.- ',·. ·.•; '\•-; ',-.• - 17 - At point 1: the rate of inflow is Q, ' the piezometric head is hl ' the hydraulic gradient is I1 · ·At point 2: the .rate of outflow is Q:,, the piezometric head is h2 ' the hydraulic gradient is :.:,. Ji ·. If the 'coefficient of transmissibility 0 the sand. is; .T' :the INFLOW a,t point 1 is Q1 = TI \'/, .the OUTFLOW at point 2 is Q:,. = T~2 W. ·...... ·Since INFLOW;. OUrFLOW = O, ' .TW(I 1 I2 ~ = o; 'and . :JI I1 - I2 ;= o;-····· · Divide both sicj.es of .the last expreslion by L, the length of the flow-p~th in the small box;. (l). I1 - I2 ok 0. L r : • I . Hydraulic gradients, r;. and I,, are expressed. in mathematical symbols as: I . . . -ilh- Il = __1_.' ' ax I -ilh2 I2 = __I_ ax ! • The terms on the right .are partial dJrivatiyes. (slop,es). Now, substitute I :·. . , these terms 'in equation (1) .to obtain . :.. . . ( ' . ' ilx ilx "' 0. L '.,' \ \ - 18 I I I When L becomes very tiny, we have th~ second partial derivative which 'is,. I I in symbol form, I ' o!Q ohJ •t- lim ox o2h L->0 ~ = L ox" • \ I The expression o2 h -= 0 ox" I 'is Laplace 1 s equation for flow in one dim,nsion. Now consider another wa~er-level conlour map . . .n.:i-/ ~/' . '\_ '· •' . The.flow is now in ctionS ;-__north and east~ Water enters .. the small box on the south and west north and east sides. By using a pr,;c~ss similar to obtaining L equation for 'flow in· one di~: ., . \• . . ·'. . . . I mens'ion we can.~.ob. tain LapJ.a~.~ 's :.equation fol r flow in 2. dimensions: .. ' . ·.· .....·. ·'o2 h o2 h . . . . ox 2 + ol j= o.. . . Suppose now that water also flows dolwar.d (in the z~direction). in which: •::: :·::.:• .:::};:::;~·: -po~ot oe now. ,.,,.,, ·;. "~':=. eo: . ",..·. ...~=·o:·;·: '··'.' ' . , ox? .,_, . '.• ·, .. ()2. ·;:;< ;,- '. . ,, )~ ·:, .. ,::': .. ·"'·'· ., ·,: . o( . '• .'X'·.... ~. ·.~ ...... ' ...... "1•. . ·-· . .. -.':. ·.: . ';·~- ,;, ; •,'{;· ., '..' '• ,: ,,• ·,_·. '~ . - 19 ,. > Laplace's e~uation appears in terser symbolic form: . 2 0::3h ·-· ,. \1 h(x) = o.J' = 0 ...a o2 h o h v-h(x,y) =-+- = 0 _ o.J' _ a:}" o2 h 2l 2 h V'h(x,y,z) =--+-- o.J' a:/ 2 The symbol \1 is the Laplacian operator. The important thing to remember is t at whenever a fluid system satisfies~ ( ,.' . Laplace's partial differential equation ( teady state equation), then at an':,:·. place in the system the rate of inflow eq als 'the 'rate of outflow, ·and there is· no change at any time in the amount of fl id stored in the system. Furthermore,·· the hydraulic head at any point does not ihange wi ti1 the passage of. tiine ,- and:~ ·,.,: there are no fluid losses or gains within t"ti1e syscem. _Returning to the water level contour maps recall that we considered only '>.the in-cernal par-c. of a flow systemq Becau e there was a ·flow conaition fluid -/· obviously had to enter and leave the· systeb at places external to iL These·' places are called boundaries' and ti1e description of the fluid system becomes ·'. " ·more meaningful if the location of boundarres _and the piezometric head or oth_e_r: , ' . :cond.itions--the~e are. designated., Hence, a flow system which is described by a_··· · i-.... ::~ part~al_ differ~ntial_ equation and bonnda!-y conditions is·· sometimes C!llled. ~ .. · •·, boundary_ value problem. . - ,.· ;. '. ·_.Laplace's equation is generally of lirtle· benef~t in our quantitative·.· ground-water studies., Its greatest utilitr is in problems of drainage and .:see.page- where steady-_state flow conditiohs occur. In quantitative ground- .: ; .Water studies, ._We are USUally interested ihnon-steady flow, >Therein piezometric : ':.he_a~ _(or w~;er ?evel) a~ ~7point in a syrtem ch~nges with p~ssag: ofti.~e_(' ~~d··• con~equeritly·: storage' of water :in ·the syste c<;mtinuously changes. A simple_::·<_;·.· ·. ' ;_._·' . . ' ' :~ ·" '' ., " . :-~-~ . _, - : ~ ~ ' ,:j: ·· .. - 20 ... :. I general equation describing non-steady flow! is· . INFLOW - OUTFLOW = CHANGE OF STORAGE. I Think about this for a few minutes. It is~'ndeed logical and is applicable. not '\· o"nlY. to ground-water systems but also to s face-water reservoirs. ' We alter this equation slightly to consider passage of time: RATE OF INFLOW - RATE OF OUTFLOW ~ RATE OF CHANGE OF STORAGE. \ Now let's return to· Laplace's equation in two dimensions, I o2 h o2 h I \ oi' + oi' =I o. ·.. ·, \ . I ·'Recall that we obtained this after dividinf an expression by the coeffh!ient' of· tra'nsmissibility;. T. Then it is permissible to multiply both sides of thi~ eqU:~ tion by T: 2 2 m ~ o h o hl\ l \. ai' + ay" Jl= o. ,''"'' " 'm" meom"""' ''""''"' tlllit ,+""' w' ootc>o: '"'" m tdenht<>_:: and the left: side of the equation. is J.n race a Scatemenc .OI. the .difference ·.in · .. ·,. :· i · rates of inflow and outflow. ·But in the non-steady state the diffe:r1ence '0etween the rate of inflow and· . ' ~, . . .' .. outflow· will riot·be .0 but 'will be ., .,·. 2 2 . T ( .Cl h + il .h '\ = RATE 0 CHANGE OF STORAGE •. 1 · oi' ai") To express:rate_ of change of storage thematically, first consider the . . . . . ' ~. . . . ' . aquifer Pl'Operty of' storage. 'coeffiCient o storage:,. S ~ is defined· aS :.the· volume •. ( ..·'. -- . t· • . . . '• '· '. ·;·, . ;change of<"'waterCin storag :. · hydraulfc li~~d;.;··or: -~ ..• :; ·: :;_:.,• ; . J 4 • . ' 0 :· ''::: ·, '•: '. .· i.~-' .. ·.·. <&'::;'' ;_ .. ··. :,::~::~.... CHANGE· OF .·STORAGE' = ( ;:~r. GE OF HEAD) X 8, ·. ,•. ' . '.-; .• .f RA.TE ·oF CHANGE OF STORAGE = OF. CHANGE OF HEAD ) . x S • • ·.. ·, . <'·;t":." / " ';. ... '• ., •,,• -~- - ... . l .. , ·(- .. {' . . '· .,; - 21 - Now (4) RATE OF CHANGE OF HEAJ:i = ah , at the first partial .derivative .of head with lespect to.time. · Combining equations . ( 2), ( 3) and ( 4) '1 2 2 T -+-o h o h) s ( ah) ·c · at· . ox" a;/ ·.;., Divide both sides by T, · This is ·the non-steady state flow equ tion in 2 dimensions . . Using Laplacian operator notation,_·-. ~ cF)-. The non-steady state equation is commclnly calleC. the continuity equation, or the diffusion equation,· so named because of .it·s application to heat flow in solids. A.system which satisfies the non-steady' state, or continuity equation is I called a transient system because changes in the system with time are considered. • I It may be of interest to note that rr.athematicians c.lassify the Laplace and· i i , the. diffusion equations respectively as elliptic and parabolic types of partial . differential equations. The classificationI is based on similarity of appearance I of differential equations and algebraic eqJations of analytic geometry. The I ., . . I . classication does not reflect shapes in fl,w patterns or bounda~ies ... '. ~--.. ·" In summary the two important difr'eren,ial equations .we have ~O"':.sidered· for· .water-resources studies are: i1 , .. l. Laplace's equation (steady st,te equation);:·~. ' .. 2 ~- .. · a"h a h I " ,,·· -- +-- = o, ',• '.\. ax" (ly" I '. \. which specifies. that anywhere in the flow system . . .. I . . • a. RATE OF INF:WW = RATE OF OUTF:WW '· .' . ,: .. ·- 22 - ...; I :•. b. no ·change in storage occis . .... 2';_·:· The continuity equation (non1steady state equation), 2 2 2l h + Cl h = ~ c(lh )' ax" ilf T at , which specifies that . I RATE OF ll'IFLOW RATE OF OUTFLOWJ = RATE OF CHANGE OF STORAGE. It is strongly emphasized that the o,rm ·fluid movement used herein refers -to bulk rate of flow, such as gallons per minute or cubic feet per second;. it does not mean the discreet velocity of .in ividual fluid particles such. as feet per·day. The reason ·for this distinction will be shown.iri the following chapter which describes the expression 'of systems with. edectric~anal~g "; . ·"' ,'' . ,; / p- •.· :·,. . , :);>' , .. ',·I • '.S •' ' . ·,,. .:: ·:, .· .,_ . . ~- . ,, .. ;'; ' ; ~.:< ,' ..... _.,. ·-·-~ ' ·-- .' ,; ... '.· ·,' .. ·!"·.·.. ,;·;! ,:; ,· ' ; ··~ ~-. ( ·• t-'l,<''i "' ····· __.. . "-:.-;:-:---·r '~- " ·." _,.,. ' ~- ··. . . -~-: •\'' }, ' "· . ;·. >'' t:; ·"· ., : .-,< ' . ,. . ~-· •1'.;. ': ~-. .\' 1. ._,. ·,.' '.;,> \ '.· "'• ,\•, ._iJ,· ...... ,:~.. . - . ·' i ~.;.. . . '· ! ·.' '.--~ ·--·· '' :.. ;. . '. 0 • • •• _., ~ ' -..... ' ...... - 23 ',,. ·' -, .,_,· .•.. I I \· '\ Chapter V FORMUlATION OF HYDROLOGIC PROBLEMS INTO I ELECTRIC-ANALO& MODELS This chapter presents an explanation 01 f some oi" the analogous. properties.· '· \ I of flow and storage in fluid and electric kystems .. First,· considering ·flow properties, wl begin with a special case of fluid· . -· ' ', . . .\' ·flow through a pipe. ' . .. - - - . f - - h ' -- RESERVOIR i 1-W--11 \ ' -- l'>w ~j_ D ~ ·. \ . .. ·-,-. .. ' \,, \\\ Fluid flows from left tq _righ-r, throug the square gravel-fil'led. p i pe 'shown :. . . .. ·:;;: bec~use "there l.S a· O.lfference· :w pressure from lef"t_ -co r1gn-c, along one flow· . . .path; . water flows in the d.irectior,. of leas-6 pressure.. The rate of flow, Q, _.:depends in part on the difference in press' re or piezometric head, h. The gravel~ filled pipe offers resist8.~ce· tO flow;. the amount of resistance depe:h.ds on cr6;:;s .. ·.. c• . . - .· . . . . ·.sectional area;· length, and. ~:_rmeability o4 the gravel. The more resistance . .. ~ . ._.... : . ·.' . f . , there Is to· flo)' the le'ss, the. rate of flow will be; :·The. ·rate of· flow thrsugh;; .· ··.:- . '. '" ' ''• ., ' . ,'' ... ~--the pipe can :be, exp~essed by the following equation: •.. __,_ ' . ·.. ~- -~ . .., :.,-:;: . 'l,• :-· ,. '_, ' ' ; ':' .•· '' '' ,• ;·· ,,. ,. .·, ·.: ' .. ·· .. .. '· ' . : . .._, -; ,, "~ ' - 24 : .... ~ _.., :·. I ,;her.e .R is the rate of flo,; through the pipe, I P is aquifer coefficient of permeability_ (applied to the gravel -in the ·pipe), T is aquifer coefficient of transmd sibility (a~plied to the gravel in· I . the pipe), and ! I Y is hydraulic conductance (Y = TWlL). --, In words, I RATE OF FLOW = HYDRAULIC CONDUCTANCE X HEAD DIFFERENCE. I Next let us compare the corresponding properties of flow of electricity. , I 7- through a rod, or resistor, as shown -oy th"e following illustrations and scheffiatic diagram:_ + .· ... Batt~ ·-'' I Res is lor ROD (resisior) , Similar to the flo,; of fluid in a piJe, electric charges (coulombs) flow _\ . - I from lefc to right through the rod becaus, there is a difference in electric _"pressure, or potential, between the ends If the rod. The direction of flow is',· .. .·, frorr higher: to. lower potential, and the mire resistance offered by the rod to the 'flow of charges the less ,;ill be the ate of flow,· L ·rhe rate _of flow of electric cha:rg.eS·. i~_· ~CUr~erlt·,: :me-~sured in ·peres,·. or.- cololllllDs_ per ·second~· ..The :·,appropriate equat~on of 'no~-:of-,eledrici y through therod i'~ Ohm's Law: l I=RV:dK:V:; where "".' ·,.;;. ,---:-- · · r' .= current, in- amperes,,· v ='di:ffe~~nce in ele~tii~ potentia ' in .volts (between the ends of. the-rod) .•·. ' . '·.'- " ' ;:.· -·~ ·. ' ., ~. " ,'1' . ,•_. -~ 25 •"' ' . ; ... ' ., - ,.: < 0 ,·, ·~. \·,R = Corresponding ana-logies between the fluid an<;i electric, systems dis,cussed:· are:, ., ,-_presented below. 'I'·. . £ . • t .. Fluid (Q '= Yh) Electric . (I = KV) h, hydra~lic potential or 'piezometri , V ,- electric potential, in· volts head, in feet .·.... Y, hydraulic conductance, in gpd/ft K, electric conductance, in. mhos .: : Q,'volume rate' of flow, in gpd I, -electric current (rate of flow .- of electric charges); in.'ampere's ·_,"· -With these corresponding properties one can use an e-l~ctric anal~g,in o~der': ,, : 0 I ,,._ -·-to obtain fluid-flow-rate.- _I , , _; - .. . ~~ . Q = c KV ': ' .- • I where c~ is a scaling factor or constant of proportio~ality. I '0 Another flow. property common to both fluid- and electric sy~tems, is, illus:.:: , ,,_ -~·,_. "< -~·< ·'\'• ·-·:\' .:, t.r~te,~ .below: ~~ "'· ,, . \ . '' :::~ '<, ~ • • • ,. • ' . -~- .' ; f '' •' '. ' ., : •" ''!•' . " ~~ .. ,:\_, ': ·. ' .,.:. 0 _· ... . ~·: "'"'<.' ~, : . . ~ ..· • 0 ·./.· '. :- _; ; . ,•, .,·, .;.. , : :·,·: .... ,· .: : .. ~ " II .. '-- 1:; - ~ I • ., " .. --0 ' ,,.- lz f " -. "- .. ~ .. ,The electric system shown on the illustrates Kirchoff's Current Law, -.:·which states· that .the sum of •Currents flow'ng int~ ·a· jur;ction equais- the silln -of 0 ° 0 0 0 .. ·,. _.. ,' 0 . .'' currents il'owing -~;,t, of 0 the junction. .,._. ::··· v :-' ·- . : ·-·:,. •.··. ~, ' - 26 An important property of. f.luid f.low ich does not have a corresponding· analog :Ln electric flow is that of velocity. A fluid particle can have practically any velocity, but the is the speed of light •. Therefore only rate of flow,, not particle velocity,.· in fluid syst"enis can be simulated electric systems • . _, . systems, and begin with fluid systems, Consider the diagram below, __ Jn~hich fl0ws out of a large reservoir ,into a "small storage tank when a valve opened. -:. STORAGE TANK, "' RESERVOIR '._., ... ___ , When the valve is open·eo.. fluid flows to 'the storage t'ank and the fluid level in it rises. The amoun~ of fluid in \the storage tank at any time after " i "the valve is opened depends upon the the tank and the height, h, of the fluid level in it. It is intuitive that rate.of filling is greater at ~rst· \.When the difference in head between the and the :Cank is greater;_ or rate of flow, Q, through the connecting decreases as the fluid level in - the tank moves-closer to the fluid level the reservoir. Furthermore; as, )1 \ \ ... - \increases, the amount of fluid stored in tank increases. - An 'equation whi_ch ·expresses\ ' ·the rate·' of change of storage ,"\' . \ '· RATE OF, CHANGE OF: STORAGE =- A'" X OF CHANGE OF HEAD) = 27 I \ where A is the cross sectional area of th I tank. In symbol form, Q '=A~. at! If the tank were. filled with sand or gravel having coefficient of st'or~ge; . S, ·then . .. , Q = AS h • . ... t "': Similar properties occur in electric storage. Consider the sketch below. '-'·and the accompanying schematic diagram • . . ~... '--·' ,.,.·. .··- ,_ ,. , ... . --· :t:·· .. .,.-- .. '· ...... ·.-. Capacit.or Capocilo( · When_ the switch is tlirned on, positive el ctric charges flow onto the capacitor plate on the left, because of the_ attraction by the_negative charges on the other ,_;_plate. As the positive chargesaccumulat~ on the left plate there is a posit:lve':~: · , __ . potential buildup; that is the positive Jltage voltage, V, on the left plate .._: becomes _increasingly greater. in time, app_ oaching the potential of the positive. 1 ' I battery-terminal. At any instant, the am~'unt of charge stored ~epends on th~ area of· the plate (capa.citance) and its e ectric potential, V. However as ·the-·· \· ,. '•" potential increase~, the charges ac~umula~e .at progressively slower· rates· since' ·• . the attraction owing to' negative charges dn the right plate is prog:cessivelf · !'. :: ri~•,:·:::.,,, '""'"'= coi e>ecte • .. ~ • •• ~ • . • ; ~·· ': ! ,., ------~-----·· , __ ---·-··· -·· ...,.. .•, ;' ''; : •W ' . :.·. . ;; or ..' ',,. ,··:' ,,·. '·.. , _,':where 'C ,,is' capacitance," in farads . . , . ~"' ,--,,: :·,·w~. ·now•,di:Jmpare analogous storage prop rties in fluid and electric systems. · ·_r In the fluid flow, system described, the raie of change of storage was Q, in ·gal-< lons per day. In the electric system described, the rate of change of, storage, . was I, in amperes, which is also the rate Jf change of storage of charge. A < I < < capacitor stores electrical charges' in a ,nner' similar to that in,which a ,eon~, ~ tainer stores fluid. Hence, Q is analogou~ to I,, and AS is analogous to C,~ :, ',, 'A capacitor, ';then can be used to simula'te lquifer storag~ properties, Using 'the "above rela,tio~ships, an equation: for an eltctric simulator of fluid storage ··'· , · would be ··- •- j . , RATE OF CHANGE OF FLUID I STORAGE = c, ~ ~~' , " . •'. ·· .. AS oh =n _,av at ~. at ·· where c, is a scaling factor. It may be helpful at'this :point to retiew previous d.iscussions. First·, : :,::.·_, ... ' < ~~.;.fluid-flow fielcis and the cor:respon6.ing partial diffe:cential .equations wh~·ch :. ·.. I ',. , : describe these field .were ·explained. Then fluid. flo"' in pipes was compare;:d~ t? ·. ::electric flow in conductors (or resistors) and fluid storage ·was compared to, . ~; ; electric storage. The problem of how to present a fluid-fl '. express the flow fieid as a pipe network, nd then to repr-esent the' flow of · ·water through the pipe network by the flow of electricity thr-ough a netw~rk 6f . . · ,:electpie components, To demonstrate this, ,consider a stead -state, flow system, .or aquifer, ,·which ,...... ',is describable'by,Laplace's.equation and b undary conditions. If. th~, flow: field, - 29 . i. '··is an aquifer it might be like th~s. , ..'· Now suppose a pipe network is con such that boundary conditions, head, and general direction of·movement pipe network are about the same' . as in the flow field: 'The. pipe network a'finite difference app~oximation of 'the:. flow field, ,.·r . ~ . :·.; ... ., . •··- . -·-,- 30 - The pipe-networK system·'of ,fluid flow can be modeled Hith a resistor- . network system of electric flow . . . .·· '' .>i_ ~ t,Pjf{,c, ,; and since head,, h, is analogous to a similar steady-state flow .. ' . ~~- ', ' ., ~.:equation for electricity intuitively follo s: . .' ·': o2 V ?/V . '·I; To oOO-c mlueo o< ~He;;:~:,:: ,j,:;,,, model.loCo nlueo ." "';'• ; .• - ".there must ·.be some scale fac"t~r c,, such th1t h = en V .. .t!tp:>'tner.. nore ,,._res::;..s ~.oaD.ce values· i~ the model must -be sucn tha·c 1/R = c11Y where c1 is a scale factor selected on the lbasis of curren't-voltage capabi:i.ities ·· of the electric eqt,;ipment used. In order ~o obtain values of head by using an. electi;"ic-analo~ model it is ::1ece.ssa::cy to l~~asure voltage at. each jllnc~ion or -~:node .in the model, and multiply the voltag1 obtain~d by, t~~ ~calin~ :actor, c, . . Now· consider the case in which a groUIJd-water system· ls descrlbed~ bY: ;he 1 ,.:diffusion., :ar·non-steady.. state, equation, \ ···· ·:• ·. • · ., ··' · , .. ·. •.• ~:~ + ~~~ ~ ~ ~~ • . .... ' ~·-~~.:...._~ .. ~.·:_._. ___ -:.._.. ~•.:,... .. ·,.::. ·_...:_, .. ...;. 31 - I \,. '"" ,.. .. . ,,,,.. ,. ."'· .. ,. "' "'" ,·I '". '"""'" . """ " .. ."""" . that rates of inflow differ from the rates rf outflow. An electric-analog model . .rna~ be constructed similar to the steady strte·system except that in addition· I .ther"' is a provision for storage at .each node, a capacitor. I I +I ·:. ~ lndica!es "grou~d" -~ : ' . ·. ,• .. The an~logous' electric nci~~steady state equ tion is ... : . :' ... } . ' .~ ' . ,. ,. . ,, . . ;, . where ' ,, .. .l - . ,.. ' . . t. --- . · .. ,. ,, \.no.' . ~; -.. : ;, . · "·· .. c, ·is: a scale· factor,-· )'·i <:: . ·--~ _,_;,-_ .. A .is.land·surface. area represented by a node, . ' : ~ ·> S is -~qU:ife~ .coefficient. of, storage. ';: :-,\gain .h ~ 6h V; however, since this is Itr·a.· nsient system both head and hi: .··anaiog, vol.t?~~ (potential) in any place in the system will be continuously. ·_:~, -·." .. , ··- :·:~hanglrig~-: ·Hence a_ measurement· of head. or·v-ltage obtained has meaning only if ·.·it has &. time reference, and. therefore ir. ole trans1ent case a provision for the graphic recording of voltage with. time is nlcessary. A non-steady flow co~dition in 'an aqui er is usually the. result.' of pumpage :from ,-;ells .. Pumpage. ~an be· simul~ted in an electrico.analog model by extracting - 32 - c:re.:t· frol)l'.the model at the appropriate lodes. The. variations in rates of extraction'~~: current corresponding to timJ-ch~nges in pumping rates are ac-· : complis~ed by the ·application of current-gJnerating devices. . Although electric-~nalog models desc~4bed have been for two-dimensional flow' resistance-capacitance networks also [can be constructed to simulate .three~ . I .. · · . dimensional flow. The following diagram 1llustrates connect1ons of resistors at a node in a three-d.ilJ!ensional steady-state flow field .. '•: z ·' .-· .. , . ' ' .-. i.- '·' ·,', ··,' ' i .. . ,, ''· i.• Electric-analog models are advan'tageous in 'the study of flow problems in~.·~ ;: :;volving non-uniform permeability and irre11larly-shaped boundaries. , ,. . .. , ·The electric-analog models discussed ave been the passive-elemen't type.' ·~ "'- \, "'.T;;;c;;,.;,:s·e·· no elements in the models generate, electric energy. If the contrary ',were,.· true,. as would be the case if electronic amplifiers were incorporated .in a then the model wouldbe an type, 'Active elements ,:.;:·:;·~model, < . ·. · ... ". ,.... , active-e~ementI .. ··.• .. ar~·u::d;'•. in' generid purpose ~analog< computers (elect diffe}ential' analyzers) which)'':.~ - .• . . . ' " . ._: . . -~ . ~.: .. " ' ''·' .. · \ -~· .. : .': . .' -; ..... ,. .~ ; '·' ·- ~: ,' •.t, ,...... _ . , ·-· .- ,j.'' .. .-· · .. ·: '•,. <'"!•' •;, ( '• h( <.;t ' ~ ,.. -... ' ~-, . " .. > '. ~ ·,·, "· . ,•' . ' . ,_. •.'. . . ·_,- ' ~- ·... ·-·.' ._., ,, ... · 33 '. .•.· ·,,,, ., .. ·------···. -- -~ . . - Chapter VI ANALOG ·COMPUTERS (Electronic Differerttial Analyzers) .... ' Analog computers are general· purpose devices· which employ laws of physics·· for performing mathematical operations. In example familiar to· many is the slide. rule. In this chapter ari analog co puter means specifically the electronic· differential analyzer, so-named because o its application in solving ordinary ·· · differentiai equations. Analog computers are simi liar in construction to. the electric-.analog models·.discussed i!J the p,evious chapter, However, an electric-'' analog model is'constructed for a specifi1 problem and is not· capable of handling· a wide variety of problems. The analog computer has 'the same. elements as. the resistance-capacitance . . . analog model and many more. However in tJe analog compu'ter, componen'ts ~re no't ., .. permanently connect,ed to one another, butjare wired into a cen'tral patch board .: . - 1 ,, unit where they are easily· connec·ced 'toge her in any desired combination. ·•.·· Although·an analog computer can be oJerated as a direct 'electric-ar.alog model {passive element), it generally useJ.amplifiers (active elements) . and a variety .of other electronfc.. multiplying voltages. Mathematical integ+tion problems· (differential· ,' . eqliati~~s) can be· solved:by:using·capacitors in accordance with. the following.law: .. , . ·• ,· - A I .. .:' ; ;· .,,' . :'.. , ' ';.', ·K / ...... v~ ~·~ ~dt; :,< ,·: .i: • ' " . ' ... i ' ; ., .•.• \ ,, ~' .;; '. ,,\ .. ·· •" ·. -,_.· . . ,., .... _. ' ' ·. :· .. ~- . . , ,-. r· •· .. · ., ·.·•_ . '. - 34 - This means that if there is a variable flow into a capacitor which might ... '· , .. graph· as· follo.ws: ., . -· \ . ·-~ { ,· '' . . !,. ,y ·.: . . , :;, 0 Time (I) A ",: them 'the voltage on the capacitor at time t = A, would be the area shaded under·. the curve. Hence, integration, a procedure, is a method of obtaining ·• ! the area under .a curve. Integration is niEttloeJnatical process of summation but c> the process is physically simulateu C\.trrent flows into a capacitor.·. ' ' The principal utility of this process. is the solution of ordinary dif~: t . , ·:, ferential equations, which is cione by Differing slightly from the ·.:,J-. '~--,.:~:partial differential equatiO!l 9 an ·. ·~: ther'e are. only two· varia~les; ir.L an One of the principal differences the a.nal_og Computers and. the ari~lo'g . ,. ~ ~ '\ models discussed. is that computers amplifiers which compensate · for electric energy losses. Hence energy can be generated in the·, system, which clas$ifies an analog as an active element analog~ · Con-· in the previous chapter was a passive. .. r . ' ~'-element ·system, because no electrical was· generated in the. ~ystem. '· · An' analog computer has one 'or more phic recording devices'which can.be' "' ' connected to· any· par't of the system to voltage. variation; with res~~ct ·: ., ' . ' -~ .' . ." ,·__ , :'r ·:.to time .. '' ' . - 35" ,·,' In order to solve a partial different"al equation with an analog computer, the equation must be expressed as an·ordin ry differential equation. For example, :flow"in an aquifer ·can be expressedas_a s stem-~f non-steady.stateequations:_:. :in whichj;he Laplacian portion of each equ tion (the left side) is s;t in _a. ,.. fixed finite-.difference form; The only otler partial derivative remaining in •.. ea__ch equati~~: is oh/ot ." Each equation is hen an ordinary differential equation,::. ·'<' . '/. ' • . ' .which elm be -:sc;lved. by the computer. An e ample of solution is presented _in the ._.,.' ;"·:· f.,, foliowin,(cha'pter. · : . . . ' ' . . ' .. " . '. ;i ... ,. . ... ''i ,. _, .'< •' .,.·· ,._ . ·. :.. '." J:i ,. ·r "I' ;_ . ."' r: '. ·., I."·· ;,, ::~·:.~·~_-.. t;.. J .. ,.·_. •, i -. ·.. T ·-,·: . ~ .. .'~~->--. ···- . :' .' ~ .,,, '.•. { . ; ...... \,'.' :·· . •.'' .... _.,· .. --~- ._,_ I. <',"·" ~ '· . :•'· ,, ';'. . .. . ' . ,_. ~. • r .. ,. ,,_; ' .~ . ·'• . .,_' . . "--~ -.. _,_ .. '_,. •-.. · ;'•; •'. ... ··'· f• ' ; . ~- ~ ,. •;'-. ·.;., :, .'. ~- ;{. }.:' .' ~; ·' ,, ; .,' ·.;-· .. ·.·. .,, .·, ... ·-. :-" ,._ .. _,. ·-~ .. ·-., • -!: - 36 - \ \ '' Chapter II EXAMPLES OF SOLUTIONS OF ROLOGIC PROBLEMS WITH THE USE OF DIGITAL AND LOG TECHNIQUES This chapter presents. selected cases }rom the literature in order to show .how computers and models ha~e been applied I to hydrologic problems, resulting .in a· considerable savings in man-hours of wor which otherwise would have been re- · .quired had these problems been approached hrough manual.computational methods. D1gi tal Com ute Methods .. ,• ' ' . ';·, ''( . i' .. Perhaps the most widely used equation iri quantitative ground-water ana1ysis· :.:.is the non~ steady 'state e~uation d.escribin radial flow to a discharging.~eii. '' .· ___,_.; .·. ·.' . ~ ' .... • • '~. ~- ·. i •• 'ThiS :.eq~~ti.On is commonly )morro as Theis' on-equilio:l:'iUJJl. fo.rmula: ,, . · ~:- . ' ' .: ' -~ ., ..· • j ~ ··_-.. }'\ fi'' ...... ,, .., ·--~'" ···.-·: -··· r 2 S . · .. where·u = 'L87--• • ..•. '::· . tT ,. .. '. ,· :and.:··: .r = distance from pumped well, in felt'. - s = drawdown of water leve_l, in f'eet 'I .. / .. ,: t·=. time since pumping began, in dayj, .· ',_, Q = pumping rate, in gallons per minute, ···.-S. = coefficient of· storage, dimensiorless, . J: - . ?- ' ... '' :· . ' .: -~ .... f ': .:...... ~-· . . •'• :. '.: ·· .• ·. . -· . '· . ' •, . . "'·' ., ..... ·';,• This formula determines the drawdown that would occur at any distance from a pumped well at any time after pumping began if values of coefficient of storage, . . . I . coefficient of transmissibility, and pump:i!ng rate are known. Because the I equation is tedious to solve, tables have lbeen compiled to aid in its evaluation. Nevertheless, if very many values of drawdown are to be obtained the procedure is quite time consuming. A digital comput:er method of obtaining des~red ·answers . is used _by this agency for aquifers with r echarge boundaries. The program .also 1 provides for automatic machine plotting ofl results onto a graph. A similar program solves the equation for a system of arbitrarily spaced wells which may: <#- have complex pumping histories. \.: \ The California Department of Water Resources used a digital computer in ·the Los Angeles.Coastal Plain area to the effects on water ·levels as anal~zeI . '\ .. a result of possible future condition·s of bumping and artificial recl'large. ,. I \Prior to this analysis the aquifer charactlristics had been obtained by syitthe- ·sis using an analog compu'ter. In using the digital method for analysis the 'aquifer was 'treated as a flow system which satisfies the equation vf continui~ ·.·.For computational purposes, the aquifer was hypothetically modelled as an , . I '\electric network problem. The aigital .computer solved the 'problem for water :. 1~\el at each node by a system of simultanrlous differential equations expressed in finite difference form. The ·oasic equa ion at each represent·ative node for ' ... . . ' a given time increment ~t is .. ' \ \ . i. ~·--·' ·. . . SUBSURFACE INFLOW - : SUBSURFACE OUTFLOW . + RECHARGE ·' . -~ · EXTRACTIONS ·- CHANGE STOAAGE = 0. ,. ~ ·.. ·_ ... ·.·· . ·. ,~.:. .. : ., ' . ,' ':'. ' '·· . .'• .. ·. _:.-_ ...... -: ·. -~---. >.' . ~·: :· ~-- ••,: . ' .·. ._,._ ... :· : .. • J ·, . ., ' .. I . .·. . --,; : :· •. -...... t. ,. 38 .. ;i; _-.( ·. ~ .. :: .... The corresponding finite difference equat'on and explanatory diagram are shown· l,,' below. 5 .&.§._ (lil + 1 - h ")\ . I (hi~l ~~-r' Y,, -QJ+l 6t · . i=l: .. "'I Polygon .subarea = As I .4.,_--:,YV,y(.i~~9;¢~:-1Pumpoge or recharge · at .node. B = Q • •. ~ •. 3 :!" The superscripts, j and' j+l,· denote the orrered time increment applicable to the . . _.,,variable superscripted,. 1 The water levels computed (h{+ ) are bnly for the enG: of a st~p .. or increme"rit·~. . . . ,• ,' ,.. ,-- I . &; .., ..of time in the complete pumping history. fhe system of simultaneous equatio_n~ . ''must be .solved for each successive time increment, beginning with the· first I . . ( j=l). Computed future w~t.er 1~~~~; ~~", plirited out or punched. on carcis and are later used for econom1c ana..t..ys:ts.. .t . · One advantage of the di~ital computerlmethod is that computed·water-level > data can be presented on punched. cards or magnetic tape and used as input data· ,, in other digi~al -urograms. . . Another acivantlgeI is that a solution for· a p_articu.lar analysis .problem can be easily o·otaineci on Ishort notice. Digital computers are also.useful in analysis of surface-water hydrologic . . . . . I data, particularly fn the predictions of r~servoir performance. Suppose that I there are .several·possible sites for a reservoir,I and it. is desired to determine. . , • '. ·' • • • I • • - . • • •••• which location would be best .from the stanqpoint of yield and storage'character-. istics. It is of particular ··interest to lowI how· much water can be expected-- ··: 39 - . I from the reservoir during a drought. A fundamental·equation of continuity applies to surface-water reservoirs: INITIAL STORAGE + INFLOW !EVAPORATION USE SPILLS = FINA~ STORAGE. :rhe equation describes an inventory and cad be balanced by a digital computer on a daily or monthly basis depending on ajailabili ty of records. If record-ed \data from. the .worst drought are used, then a "firm yield" from the proposed reservoir can be predicted. Of particular appreciation is the immense saving_ ... -in 'man-hours of ·tedious .:omputational work which would otherwise have to be performed by engineers. I Analog Model Methods :'-:_·· Use of direct. electric~ analog models ,receded the use of analog computers· .. by many years. For example a resistance-capacitor network for analyzing fluid. flow through porous media was developed in the early 1940's, several years before. electronic analog· computers were opjational and commercially available·. :. In modeling a ground-water system wit an eleccric analog, the aquifer ' .· system is fi~s'C partiti~ned. or di v~ded incol"polygonal sub~r~as. The flow of .wat~r througn subareas _1s s1mulateo. by the rrlow of elec'Cr1c1ty through resistors -ancl the values of resistanc-es. are selected lso they represent hydraulic con- ..,~tictances of the aquifer.· Capacitors at juhctions of resistors (nodes) simulate· ·.:storage _properties of the aquifer. Values ~f coefficient of stor13.g!" and .. ·-,: J:l~dr;,_ulic conductance are obtained from pumting-test data.· . .After the model is constructed itis trsted to see if response of .the mo~el . 'conforms ·.to the ,historic response of the aquifer:.- To do this, pumpage history • ~is si~ulaf~Ci by ~ 'vary~ng electdc ~urrent rhich is i~jected into or ~xtra~te~ . · ·. from the model. at .appropriate ·nodes. Volta es at the. nodes are graph1cally : , . :~~~~rded Th:·voltages? ;,f cgur~~' simulat water level~~ in the aquifer.: ;If '.;_' .,•"'····-r· . .. .·· .-. . ' . ' .· . -·. •.. . . ' ~-~ :· .,.-- ...· . . ·, .· v.... ·.·,t~.c· ~...... :-:-.;~ :\'. - 40 - the voltage graphs do not conform to the hydrographs obtained from field measure- ments of water levels the aquifer model is modified. That is, values of re-· sistance_and capacitance in various parts· f the model are changed in.order ·· that the system may conform more closely t the historical records, assuming pumpage data are correct. This is the syn~hesis process. When the synthesis is carried out to where there is a satisfactory-agreement in response of modeL and prototype, realistic-_ .-.. values of hydraullcI conductance and storage properties . are, by application of the appropriate scaling factors, obtainable from the .... v~lues: of resi;tances and capacitances in 1he model. It should be noted that aquifer coeff'cients obtained by model synthesis may differ from those derived from pumping tests. Coefficients obtained from . · • -pumping tests reflect aquifer properties o ly in the immediate vicinity of the;:· ,.,, ' ~~pumped ·well- and -for a relatively short timJ. Therefore, aquifer parameters- ! . . . -,,>obtained from pumping tests are generally iodeled only as preli~inary approx- _ -· .• ·m.~tions, ·and are later a . model's response 'simulates with sufficient accuracy historic \¥ater-level data .. Of pa:ctic~lar int€_rest in the use of .analog ~node~s in grOund-wateT prob,l.ems · · 'is a· studY rece~tly conipleted· in·· the vic in· y of Houston. In the Houston area. -.'most "deve_lopment is from two. sand aquifers separated from each other by a rat'her ' 'thick, sandy clay layer. There is significant hydraulic connection between · these aquifers, and there is cons ide; a-ole in~e~bedded clay in the lower aquifer; .· With such a complex aquifer system compounded by withdrawals from· so many"· large-· capacity wells the estimation of future wat~r-level declines in the Houston area would be practically imp~ssible with manual computational methods. Coefficients bf transmissibiH'ty and storage of the aquif.;r were obtained from field pumping ' . . I ...... --te-sts a~d ~ere simul'!-ted as ·preliminary app oximations in- a resistance~ cap" . " . . ' acitance network. ·verticai permeability.of the. clay. .layer ·. betwee~. - .the .. .aquifers.··. . ·:r- .. . - 41 - finally synthesized so that its response co formed reasonably with historic water levels: The analysis phase of the Houston prob em consisted of imposing simulated · anticipated future pumping conditions and r cording the expected water-level ..declines. '' The model used for the Houston problem utilized a very short time scale, on the order of one millionth of a second o model time to one year of actual time. Although short time scaling is somet mes desirable in order to take ad-. ;,..antageof low ·v;,ltages.and small. inexpensi e ,capacitors, expensive equipJ!Ient is require<\ for simulation of pumpage. In he Houston problem pumpage was . .i '~ : . . " ·; . . " ...... ' -~ ~ simulated by several square-wave generators connected such that.their combined ' ' -·· . __ ._, ···-.. _ -~~~p~t graphed as a pumpage histogram. The simulated·water~level hydrographs .- i·· .'ati·'~ach·nbde.. w'ere presented on the screen o an oscilloscope.· In ·order to ':nave ::h:~;:::e:!·,~:-:P:::::a::dt::t:::::~::c::~:r:::. it was necessary to photograph . .,.J;' . I ine~tric-analog ·models have been used to some. extent in. surface-wa·ter work,·' ·. . I . ~ .. ·.particularly . . in. flood control. . and tidal. studies,I .and are considerably. mor~ .. . com-. plicated than those used in ground-'W'~ter prlblelils .. Applied research· :O.n .this ... • . . field has. been conducted at the ilydraulics ~·aboratory of t~e University of ·· :. California at Berkeley. . ' A surface~.water problem in California l'nvestigated with an ~lectric-ana.log .,; ·method involves two -rivers, the San Joaquin and the Sacraln:ento,. ·The Sacram~r;t6: :.. •'River drains the northern part .of the StateJ and the San Joaquin River drains. the southern part. The confluence of the rivers, about 60 miles east. of San· Francisco, is a poorly drained area charactlrized by an in;ricate pattern o_f• :·,:,.:<; interlaced waterways, called the Delta. FrJm the Delta, ·which is approximately:· at sea .level, d~ai~age is westerly .through l series of small bays into the ~ · · . '~' '' ' '); ._., . . .. San.Francisco Bay, thence'through the Golde . Gate into .the· Pacific ocean·. ·' 1.· • ·' • : '. • • . ~ ~ -~-, •. ,._. ;! ' . - 42 Because the Delta is situated so close .to sellevel and near tlie ocean, surface- · water supplies are subjected to contaminatiof by salt-wat~r intrusio~ owingto ti'de·s. Furthermore because the San Joaquin iver drains the relatively arid . and agricultural southern part of the state, it transmits poor quality water to the Delta in' summer months . . . Oceanic tides· are periodic pulsations, nd in the river and bay systems. • ·-· > from the.Delta to the Pacific Ocean these ti al pulsations generate complicated . patterns of wave reflections, some .of _which- LyI be mutally reinforcing. It is desired to make some changes in tllie channel system which will consist I of a barrier or an obstruction to the intrusion of sea water into the Delta, and the changes involved will be rather costly ald extensive. Therefore an analog .. model of the channel system would appear to ~e desirable 'in order to study •tidal effects on the channel system under prtposed m'?dification.. After a variety. 'of proposed changes are studied, optimum mod~fication plan can be. selected. · Furthermore it is. also desirable to model thl channel in order w study behav:io:, I .of floods from the San Joaquin and Sacramento Rivers so that they·can be ef-·· Ill fectively controlled. . d . . . ' Th e cons t rue t 1on an appL1cao1on or an ell ecorlc-anaLog. . - model- of a river. 1.system presents many technical difficulties, !because n9t only must channel :characteristics be modelled (anQ these charaJteristics change with depth of water . I in the. channel) but also appropriate time-de~ay devices must be built.into .the ·system, an Similar difficulties are encountered in tid.e simulation. .\' Analog Computer Methods . I The California Department of Water Resources used an analog computer (electronic differential analyzer) for aquifjr synthesis i;,. the Los Angeles · Co~~tal .Plai'n; . :The analog computer was used as an active. 'element device and . . \ ..; 43 - \ \ the correspondence of properties·· between prototype and model are not appare.t; 'Before describing the application of he analog computer to the· solution of'the ground-water problem.in the Los Anglles area, it . . . ! . . is desirable at this t'ime to· point out· some of th~_attractive f atures of the analog computer compared to an analog model. First, analog compute components are connected to a c'ent·r~l patch board, which facilitates hook-up ope ations. Moreover, the different · .. elements· used to simulate storative and tr nsmissive.characteristics of the · .•. ·aquifer are .,;ariable, hence ·adjustment~ ar easiiy performed. ·Adjustments are .·•· .. '-::._·not easil:/.".'niade. in a direct-analog model i fixed resistors and capacitors. are, . , . :;··· . '~> used to simulate the ,aqui.fer's characteris ics, in which case changes can be ''' ... ·-~"cd,e, onlyby.-·substi tution of. element~.· An lther advantag~ of the a,;alog comp~t<:r:;· '·' _,.~ . ·.··;.;:tfi'~"large ··model-time scale used. In co trast to the time scale of one micra-· · ;~c~·nd-.P~Q~~e~r in the analog model of the Houston problem a time scale of ap- ·proximately one second per year was used in the Los Angeles problem on the· .analog computer. The longer time scale alJows the output hydrographs to be recorded graphically ·oy a pen plotter· rathJ .than an oscilloscope·. Pen. plotters . I . are generally more accurate than· osc:llloscjpes of· comparable quality'~ By using: a multi-channel plotter on an analog computer, time-voltage graphs at several nodes can be obtained simultaneously. In the Los Angeles Coastal Plain probJ.:em the· aquifer was modelled by three'. ·. ·.PACE· 231-R analog computers, operating simitaneously. In order to utilize· components most effectively, ar,d model as ~ccurately as possible ohe irregular. aquifer boundaries and irregularly spaced c~enters of pumping and artifical ·re charge, ·an asymetric pattern of nodes rather than a regular square pattern was·' I .used. The differehtial equation of flow at\ each node al).d scheme.tic .diagram for:, I the. connection' ~f compo:>e.n.ts' at a node in tre computer 'are· iliustrated··:on ·~he.·''., ' following. page.·... - 44 Polygon subarea = Ae or recharge ·'· at node B.= a • ,., -,.· ... : . _.,.·,, . ' ~ ' /'.. ,. '.; ·,.' '<, .• ',•l ~ AeSe _I_ I y,,. AeSa LJ 1•11 -~ . integrator Amplifier ,..-:. ·.,. .. '.. \". \·.. . ' ...... C' .~' i" 1 ·•' ·• .::. . \ - 45 - \ I ',The actual circuit employed was modified sjightly from that shown to conserve . time in making adjustments .. However, the srhematic diagram shown is basically. correct and could• have been used. The inp11s into integrators are the subsurface flows and the surface recharge and extraction. They are integrated by accumu- . I . ~ation on a condenser. The integrator output is a voltage which varies with \ I . "'time (simulated water-level variations withl time). The integrator output is fed :into a unity-gain amplifier to reverse poltity (integrators and amplifers are 'constructed such that the polarity of the output signal is opposite of the input '· .signal). Mostiy for comparative purposes, .the CaliforriiaDepartment of Water Resources used the analog computer in the analysis of the aquifer in the Los Angeles· Coastal Plain. Although the two procedures yielded practically identical results the ;. digital coinputer appeared to be a more usefll device for aq_uifer analysis, par .. -ticularly. .because of. convenience.. "arid becauseI of savings. in . time an6. cost. . It is therefore of. interest to note that in planning for the optimum develop~ ment and management of surface- and ground- ater supplies in·the Los Angeles area bot~ analog and d.igital computers were used.. The ·analog computer was· used· . principally to define or sythesize the aqui er.. ·The digital computer. was, used:, . . '· - .. ' ' .... . thereafte? "to ana_lyz8· the eff~ctS on water G.evels that:wm.lld occ:cr from va:rious ' ' ·.' pr?poseC. schedules of pumping ami· artificia .·.--·- .. Howe:ve~:-~--~~-:thi·s .. is but .a P,art ·of the wat r_ :resources :p:r·c·ole:m in :this _area. : li:~~;:£~e_·:?_~eSen~·.: tirrie, ~uCh of the .water used in the Los .Ang·e~es area- i_s ob~B.in'e_d· :' .·. 1-rom·-~· distant·. ..:~~fa-~e-~~ter soui"ces whi~h ar , d.eveloped in conjunct-ion ~Ni th local. grou;i~-~a~;~~J,~upplies. Moreover, it is anttcipated that additional ~ater will soon be 'imported from the Feather River Project, originating in the northern part of the state; Therefore, the most optlmum plan of developme'nt of available· I .· surface~ and ground"water .resources' should t·· analyzed. Analysis of such infor- mation requires the handling of a: large vol e of statistical data and data 46 ~ .concerned with costs, available storage, f ture pumping demands, and so on .. Processing of such .volumes of information choosing the optimum plan of water· .d<1velopment ·suggest the .use of a digital Many interrelated programs . ··'· are therefore utilized in Order to achieve an efficient.plan Of Water management: ·------ ,. .., . '• '' \ ,.·~ ' . 'l ' .. ,' '. . :.·· ·. .. . '•c • ' .. .. ~ ' :,. '·". ' .' ' '. ,, ,. '. ' ' ' ', ~ • .. ' : ' ., ~ .,.. \ _., ...- ... - '.• .. ' v . ,._.-. 'I ' .:. I • l -_. '"'' I .l . ' '· \. \ I. \ "> .;.. -· ·.- .- .I '• • .'I!, I ,. '• .·. • 47 ~ .· CONCLUS ON A decision to. solve a: hydrologic pro lem through one or. more of· the ·tech~. niques of digital computer,. analog compute or analog model should. be based ·prima- <<' .· f·rily on whether. a necessity exists for obt ining the solution sought. If the . ·need does exist, then available pertinent ata should be collected and carefully: ~o<: ~ ., ' •. scr~ti~ized 'to determine (l) th~ probabiliry of obtaining reasonable: results and' . ( 2) •which: technique is most practical. _.. ·- One wa'y t;O d.ecide on the toecnnique ·co be used is 'GO personally interrogate ~ _·;"' ol-f' . ·· · various experienced individuals. Perusal • published literature is recommerideO:, .:but it must be rememoered that toecrmical ptper.s often contain; some pe!'sonal. ";.'; ·: ., vanity; . That is ,many wrHers are usually Inclined to report favorably o.n thefr · . , :~uccesses,. dismissing the difficulties encountered in ~heit .e:eforts, .Report~.;;;·. of failures· 'seldom: appear· in th~ technical journals. · •· · .· ·• '.I·.•.. ;. ,_·; ,'• :~- f .,,_,• ' • ',!•," '· ;. .. - .. ,, ' ' ·' '. ·' l::;.'l !, : . ~ ·'. ',_;·- . '·.: .,: . · .. ·.,. '·J,.; • ';>, •>'f, "' ·'. ",' ,,_ ,.',' - :~- .I ' ' _,, .. '' ,• . ·' . ,' . ' ~. ,, ~ I '1. ' . .'' -·-, .( : .} . -'- '. ':< •·. ~ . ;_"' .. . '·> ·.': '·' ;,···, ., _, -. 48 - BIBLIOGIPHY Bear, Jacob, 1960, Scales of viscous analogy models for ground water studies:·, . Am. Soc .. Civil Engineers Proc., v. 86,lno: HY2, pt.·l, p. 11-23. -~ .. . . I . . . -:--!l Bennett, R. R., and Meyer, R. R., 1952, Geology and groundwater resources of··· the Baltimore area:· Maryland Dept. Mihes and Water Resources Bull. no. 4,· 573 p...... __ I . 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