Casagrande - FILTRACION EN PRESAS

RS

ove r the lateral

)f the lateral supshy sa m e Yet the Tormously accord shy the bank The rough the Coot oC

hydrostatic presshyation of practical under field conshyiller on Fla tbush should always be

Orig inally published in Journal 0 tlle Ntraquo England Willer Works Association

June 1937

New England Water W orks Association ORCA NIZED 18 82bull

Vol L I J une 193 7 No2

SE EPAGE THRO UGH DAIS middot

BY ARTH tm CASAGR-LWE

IRood Fm- u 19341

COTETS In troouction

B Darcys Law lor the Flow 01 Water throug h Soils

C GeQeral Di1TerentilLl Equation for the flow oC Water through HomogeneolJ Soi[

D Forchheimera G raphical Solution

E ~page through Dams Ceneru C009ideratioos

F Seepage through Homo~neous Isotropic Earth Dams amp Approxima tA Solution for 0lt30 h Approxirrute Solut ion for agt 30deg Ic Solution Cor a Horizon tal Di8Chsr1~ S u rface (a = Jlt)l d ApproximatcSolutions fo r Overhsnging Di~chaTge SlITfacea (00 lt0 ltlSO) J

G ~page through Anisotropic $v iI s ( H RClrlark3 on the DesiWl of E~rth Darns and umiddote~ r 1 Seepage through Com xgtsite 5laquot ion

J Com pariso n netwetn Forchheiroer a Graphica l [e t hod Hydraulic Iode[ Test~ and the Electric -nstogy I ethotl

pp~nJ i I ll ~Feet ion of Flo Lines due to Chanlt~ in Permetb ility h Traru(er Cond itions for Li tle of ~p3ge a t ampunrlarie~ General Re~r~ c DiscMrge in to a n Verh4Q~ln l S lope J TralUl(~r Conditions for Line of ~pa~ dot Bou ndary betwe-~n Soils of Different I

Permesbility e Sin~lampr P oints in amp Flow Xet

-ppendilt II amp Graphicsl Proced1lre for Determining In terseetion betwetn Discharge Slope and

Bas ic Pam I b Compari~on betN~n HAmels Th~ore iea Solution ampnJ the Proposed Approxishy

ma(c Ietnod c Grlp bical Solution by l eC5 of the HoJogrnph

Bi bli (grlph~

ns 1

f ~ ~

296 SEEPAGE THROCGH O YS

J TR 0 0 U C1 10 N

t n t il about ten years ago the design of ear th da ms and dikes was bosed a lm os t exclusi e ly on empirical kn owled ge a nd consi13 ted largely of adopting the cross-section of successful dams Vti tb little regard to differshyences in charac t er of soil an d fo unda tion conditions At p resent we are in a period in wh ich the behavior of dams particu la rly those which have fai led is a nalysed in the li ght of modem soil m echanics T he understanding and k nowledge thus accumulating is being used as the basis for a more scien t iIie approach to t he des ign of such ear t h structu res

T he most outstanding progress in this subject rela tes to the question of seepage beneath d a ms and dikes and to the effect of see page on the stability of t hese st ructures F oundation fa ilures due to seepage com monly known as piping were for the fi rst t ime correctly expla ined by Terzaghi (1)- i

who denlo ped what m ay be termed t he m echa nics of pip ing Later Terzoghi (2 3 a nd 1) co lled_a t1 ention to the importance of t he forces c reated within earth dams and concre te dams d ue to the- percolation of shywater The p ract ical application of this in form ation h as lagged behind our understanding of thee for ces pa rtly because of t heoretical difficul ties of anal y zing p roblems of seepage I t is on ly in recen t y ean t ha t substantial progress has been m ade in the solution of prob lems of seepage and ground wa t e r flo w wi t h free or open surface of the flow t h rough anisotropic mashyterial s and of the condi tions of fl ow throug h join t pla nes of d ifferent materials

B DARCYS L W FOR THE F LOW OF YATpoundR TH ROUGH S OIls

The flow of water through soils so far as it affects the ques tion of seepa ge th rough dams follows Igtarcys empi rical la w which s ta tes that tile amou nl oLflOtL i~ diredly_proportional to the hydraulic gradient_ ThUi la w can be expres~ed ei t her in ~he form

(10 = k i or (1)

Q= kiAt

in which the sym bols have the fol lowing mem ing

(I = discharge veloci ty t k = coeffi cien t of permeabili ty -shyi = hyd raulic gradient

Q= q uan tity of wa ter A c= ares t = time

In Figure 1 t he memi ng of D arcys la w is illustrated in simple form A p rism atic or cy lind rical soil a m p l is etposlt d on the left sidt to 3 head

middot - umerAlJ reler 10 Ill bibllOCaphy1 b cd oI lW pa per

tThia mud Dot be eonl UM-d th ttw ~P-Cft m rxit)O _ plusmn in - bieh -ntio ~ ~WD 01 Toidl r

14 ToIn 01 oIJd mat ter Thbullbullnun roocitr lhMoucgt tgt toil iI rIutltd by tbe _~ niocity IW the dixhMp nloci ty dCI ~ Ibt Qlla3 titT g( lo

bull

ldps was j largely I I differshy

I we are Iieh have J~lllnding

r a more

Ilslion of stabili ty oIy known wghi (1)shy

Later I II forees IHliu of middot hi J ur middottllties of dlstantial ground ropic ma-different

JImiddot~tion of t hat the This law

(1)

Ilc form n head

middot ~ u e ol yoidJ

0 elodty

Corcys Law

for i1w Ihroui SoiI

~ 1 iOixhclIye Yeocdl

HydraulIc 6rgditnl ~~ i bull I

fl

r

FIG 1 shy DA RCY S L AII JOR F LOW THROCGR Soi LS

I ll elx ~c11 J Jy

PH~ltr tUM3

i CIjltJ

FIG 2- CZi Eltt DlfF~R poundlt T[ l QCATIOo f O R TH E FLOII Of W TER

THAOlG H HOllOGltO t1 CsOTROPI( ~ t

~ 297

293 SEEPG E THROUG H DA~S

of water hI l lld on-the rig h t 5ide to a 5mnll er be d h2bull As a resul t water gCIlIwill flo w th rough the sample a t a fa te di rectly proportional to the hydr aul ic soil

gradient i = hl -h2 bull If for example the difference in head (hl-ht ) in Fig

l ure 1 is doubled the q uantity of ecpage will a lso be doubled This linear relat ionship suggests t hat the flow of water through t he voids of most soils possesses t he characteristics of lami nar fl ow the

D a rcys law is frequently attacked as being incorrect In general these att acks are b ased on misin te rpre tation of test results or improper technique of tes ting In many cases th ey a re J ue to loss of internal stability of the

T h i ~o i l unde r t he ac tion of fl owing wate r

(SeThe read er m ay be assu red that this law is valid for the study of kno

seepage throu gh da ms equ

C GE~ERL D IFF ER EXTIL EQUHWN FOR T HE FLOW

OF WATE R THROtGH HO~OG E~EOUS SOIL

H water is percola ting throug h a homogeneous mass of soil in such a for manner tha t the vqids of t he soil are completely filled with water and no change in t he siu oC th e voids lokes place t he quant ity en tering from one or several direc tions into a small element of volume of the soil (as shown in Figure 2) m us t be equal to the amou nt of water flowing ou t on the other faces of t his ele ment of volume during any gi ven element of t ime This condition which is a statement of the fllc t that poth wa ter and soil are incomp ressible can be exp ressecl for the three-dimensional case by the follo win g equation

(2)

Th is is known as the eqlwlion of Ctm l ill uity (See Reference 5 ) In this equation Il II and w a re the three co mponents of the discharge veloci ty va

If dh represents t he hydraulic grw ient in the direction of Bow a nd rl dft dl dx dy

dh and are its t h ree cornronnls then D arcy s bw

d dllv= xshydt

can also be expressed by the oUoti ng equations

ok u= k-ax

ak v= k shy

- oy shyah

w= k shyoz

o rrly I~ ~ ~ky k-z-- It I soCgt-- 0 t prr

tio (3)

5 1

amp r bull

Figmiddot

ineu m05t

thee lique f the

ly of

I(h 3shy

Id no lone 1 own the~

Thi dare

the

(2)

I thi3 ty o

JI rfh - shye dy

(3)

CASAGRAlDE

By substitut in g Equation (3) in E q uation (2) one a rrhes at the gtnera l diiferenti llI equation fo r t he stEady fl ow of water through isotropic soils This has the fo rm of a Laplace d iffe ren tia l eq ua tion

o~h oh oh - +-+ - = 0 (4)ox oy OZ

In our problem of ceepage throu gh dams we have to deal only with the two-dim ensiona l case which is sa tisfied by the equation

ah ah- + shy = 0 (4a) axl ayZ

This Iquation r epresent tuo fa m ilitamp oj eU n damp inlerseclin al ftght anglu (See Reference 5 p 24 nnd 25) In hydro-mechan ics these cu rves a re known respectively as t he floti lines and the tqu ipolenfiallints (or lines of eqllal head)

D FORCHHEI~ERS G RfIlICAL S OLUT IO N

Although the general dillerential Equat ion (4) has been solved only for few and s imple C3Ses of seepage we can make use of certain gromet ric

rf

-C~~ ~ WNIlt null

Q-A ~ ~

417 -n

i l J3

FIG 3- FLOW pound1 Dt middot pound TH S Htl P I LE WA LL

propertie~ of fl o lines nnd Pflui potrntinllin(~ that ptTm it pflplli(al sulushytion for p ractica lly a ll tw(-d imrnsionll ltcepage problems Tlti I1Irthoo Va dp ilCd by F orr hhei nlC r (5) t( nty-fin yrar- ago

~(pLain_tbLgr_a phislJ r(t hou thl prolJI~m of d rtermini ng tile el palC h(nlll t h a shEpt pile wall -hOWll in F i urp 3 i cho PII T he gro und sll riacr is a line oi rqllJl hfJd or Iil r qu ipo tr n t iJ1 liIH the hrld bring

300 SEEPGE THROUGH DH S

i lequal to the heigh t of water s tanding a boe t he ground surface hich is h Ro

on the left side a nd zero on the right side of the wall The bottom of the penious so il stratum is a Row line incinentally the longest flow line The of sides of the sheet pi le wall and the short wid th a t the bottom of the wall exr are the shortest flo w line

If from th l infinite num ber of flow lines- possible wi thin the given hI

area we choose only a few in such ma nner tha t the sa me fraction ltlq of the a I

total seepage is passi ng between liny pair of neighboring flow lines and wi simi la rl y if we choose from the infinite number of possible equipotential IU

lines only a few in such a mnnner tha t the drop in head Jh between any pair of neigh boring equipotent ial lines is equa l to a cons tant fract ion of the IY

total loss in head h then the resulti ng flow net Figure 3 possesses the t

property tha t the ra tio of t he sides of eac h rectangle bordered by two flow th shylines and t o equi potential lines is constant (See Reference 5 p 82) If in shyall sides of one such rectangle are equ al then t he en t ire flo w ne t must canmiddot sist of squares Conerscly it can be prol~d tha t if one succeeds in plo tting co

li fl t wo se ts of cur(s so that they in terect at ri gh t angles forming squares in l

and fulfilli ng the bou nda ry conditions then one has sohed graphically tJ t

equation (4a) for this prob lem Withexperie nce this methruLcaILbeapplied in swcessfulI~tCLlhtmos t compl ica ted problems of seepage and ground aleshy H

fl ow in two dimensions including seepage with a free surface it 1

byAfte r having plo tted- ftow net-tha t ful fi ll smiddotsa tisfactorily t hese necessary in

conditions one can derive therefrom by simple computations any desired information on quant ity of seepage seepage pre ures and hydrostatic uplift For example the to tal ~eepage per unit of len gt h and per unit of l ime is de~ined from-thefolJo ~ i ng-fo rmu la~whic h [- simple to-deri~e i n

fro m Darcys Ja 1

(5)

ill wh ich 1 is the number of ~qll~rcsb eteen two neigh boring flo w li nesl

ancLnmiddotdhen umber_of sqU3res betwee n two neighbori ng eq uipo ten t ia l li nes T he maxim um hydraulic gradient on the d ischarge su rface which

influences the safet a a inst II tillin or b lows is e ua to

aI =shy

~h (6)

J

in which f is he leng th of the sO l- t -quare on the discharge su rface as th

indicated in Figure 3 a n~ ~h-Ii ~he drop in head between twO adj acent nl

equipo tential lines To assis t the begin ner in lear1 ing the graphical me thod the follogting

a r l su ggl$ tions an~ made

1 ese every opportun ity to 8luJ y the pp~ot~o~ of well~onstrucled Row oe13 SI

wheD the p icure j ~ ~ufficie Qlly abltl rbed in yQ ur mind try to uumiddot the ~ me flow net

--

1

r-

(

I

1

I

bull 1

I

J

middot 1

I I bull

I

)

~

CABAORA~OE J01

ithou t looking a t the avai lahle IOlutian repeat this until you are a ble to sketcb tbis lIo w net in a M t isCaetory man ne r

2 F our or fi ve lIow channels are usua lly suffi cient (or the first atte mpts the use o( too many flow channel~ may distract the atteot ion (rom the essential fe t ures (For e xample ampee F igupea Illb a nd cl

3 Always watch the a ppearance of the entire flow nel Do not try to adjLLSt details heCore t he eDtire flow net is app roxima tely correct

t FrequeD Uy t here are portion of a flow De t in which the flow linea should be appro xim ately straight and parallel liDe~ T he flow channels are theD about of equamp wid t h and tbe sq uares a re t herefore uniform in 8 i ~e By starting to plot t he flow net in euch an area uaumlng it to consist of st ra ight lines one CIUl laci li tate the solution

5 The lIow net in confined areas limited by par Itel bounda ries ia rr~quentty symmetrical consis t ing of curves of eltiptical shape (Fo r example see F igu re 3 )

6 T he btogin ner USUlly makes the m istAke of drawing too sharp t raru itions bemiddot t ~lIn straight and curved sectIons of 10 lines or equipot~D tial lines K~p in mi nd that all t rans itions are smooth of eltiptical or pambol ic sharpe The eire of the squares in each channel wi lt change gradually

i In general the fin t lMumption of flow channel~ will not re~ul t in a flow net coositin K throughout o( squares T he drop in head he twen neighbouring equipotrntill 1ine~ corres ponding to t he arbi t rary number of tio channe ls middotilt usually not be an intege r of the to lal dr op in hea d Thu~ 9o here the flow net is ended a tow of rectangles i ll remain For usual purpoee~ this has no disadantampge a nd the laat to is taken in to coD3 ideration in coAlputations by ~timating the ratio of the sides oC t he rectangles If for the sake oC a ppearance it i8 desired to resolve the en ti re a rea into square~ tben it beltome DICeaIlIU to change the num ber of Row channels either by interpola tion or by a new stsrt One should not attempt to Corce t he che nfe into 3(juares by a djustments in the ne ighbouring are6S unless the n~lSIIJ correction is very smal l

8 Boundary condltions may intrcxluce singular ities in to the flow net which are diSCU35Cd more in dete il in ppendix I e

9 A discharge face in contac t lIfith ai r is neither a flow line nor a n equipot~ntial

li ne Therefore t he squares alonp such a boun ar) are incomple te However ~uch a bound ary m ust rulnti t he same condition as the line of ~l-epage regard ing equll dro ps in bead be t een the point middothere the equipo te ntia l lin es in terslef

10 W ben construc ting a flow net ct1t1tsin ing a r~ SUrfUl t one should ~tJ[t by ampSSum ing the d ischarge race a nd the disc lul rge poin t and thrn wlrl to ard tb e upst ream bee until the correct rela tive positions of en tmnce point and discharge loe are o ttlUleU Hence t be scate to wbich a Bow net 9itb s free su n set ~ pia ttid II Dot be knowo until bull Inrge portion o( the Bow nel i ~ lin i~hed For SEe pag~ problem wi tb free 5urflre it is prlc li~ ll y impossible to construct (l flo net to a predetemlined 5~31 e in Ii re~ 50n3h le length of time

E SpoundpoundPOE THROL GH D ~I (1E-E RAL COSltI OERTIONS

in_almost all problems concerni ng serpage bCllealh ~h fet pile_rgtl lls or thro ugh the fo undation of a QJll all bSlll nnfJJy__co nriltruw areJ lQlf]) Howshyever in the~e throllfJh an ~arth dam_or dik~ the IljJper bound(l~r ~nnoit flo w Ii~e is not knu ~n but must fi rst be found th us introducing a compl ica tion T his up pe r boundarj is a free wate r surface and will be referred to as th e line of seepage

mong the avai lable theoeticor soill tioll - for slep ge -with a free- _ su nare the reis one elSe which is of part icuI3 r impltJr tanf( in connection with our prob lem It is Kozerys solution (6) of the HOIgt along a hori zon tsl

JOl SEEP G THROUGH DAMS

impervious stra tum that continues at B gi-en point into a hori zontal peniolls IItrll tuID th us representing nn open Qorizon t 31 di charge su rface as shown in Figu re 9d ~thjs CLe all flow li res including the line 9 f

6e~age and ill egulplt2 te ntial lics a re conpound9cal parabolas wi~h poin t as t he focus

FDrJheJl1oILCQmmon p _oQlem of s~page t hrough cross-sections in whichth~dischargeslope fo nns an angle with the horizontal between 0

0

and 1800 slch as tb~o~1Ldischrge on the downstrenJIl face oC a dam or discharg-e into- an- ove rha nging slope of a ery pervious toe such as a rock

G_~ral Con d dion for r rlaquo sur~s

(~ (Jf~fO14 or (p~3llo lr~)

F iG 4 shy GE~ER L C OSOrrlO-I rOR L in o r SEEPA GE

fi ll toe one has to ulte either a g ra phical sol ution based on the cons t ruction lLllid lorLn tt or ~ Ie JPPiodml te m_th lm til-Il l ~(ll l tio n-- I n-ei the r casp one must int rod uce ce r tnin conditions that the free water surface or lin of seepage m us t al ways fulfill

The fi rst condi tion is th at t be elevation of the point of in tersection of nny eq uipoten ti al li ne vth th e line of seepage represents the bead alo ng t his equipo tential line If we cons truct s fl ow net consis t ing of squs-es then it foll ows that all int ersectio~9r equiPQtentiaIJig~s wit hJhe line of seepage mus t be equidistant in t leJertifaL di rectiolL T hese dis tances

hillustrated in Figure 4 represent the actusl dro p in head th = - betw~n

71 t

any two neighbo ri ng equi potential lingt Th~ second condi t ion ~[ers to the slope of the line of ~pa~p at th_E

point of interection With any bo~dlry as fo r elta mple a t th e points of en tra nce and d ilrhlrge and at tb e boundary li ne between different soils

~ c_

~ -shy~ ~ bull 1M

Fin

-

izontal surface line of

ltlint A

ions in veen 0deg d am or a rock

-

truction lller ca e or line

(middotrsedion middotau along quQre

he line of Jistsnces

between

-KC at the I)lt)ints of

rent soib

CASAGRUmE

Flo 5- DeFuCTloN or FLOW Ntr AT BOO-lD AII Y

OF SoIl3 OF Dl rnRfT PERlfE BI LITY

~-~~ I

CotJdiIJ1O9J fo AJI o~~ (1 I~ c sC-6~1 KJ r j 4(j flu a ) ~ _ t~4

) r)------- ----Jr ) Jr

~ -II bull gtIJ bull 0 IJ bull igt oJ OJOG- ~~i

_- 0- II-~ I) c ) )

m

FlO 6- ENTRMIC J DI~cRARGE AI4 D T AfStEt (OOlTIO S or Lla 0shy

SU C l

SEEP GE T HROUGH DAMS304

(See Figure 5) By co nid~rnt ions based on the general properties of a lIow net one c~n arrive at the conditions which mus t be ful filled at such fh poin ts of transfer In - ppend i~J are asselllbled derhations for typical car cases If for e(om pre t he downs tream face is inclined llss than or fqual to 90deg onp fi nds th at the line of seepage must be tangent to th at face at the discharge poi nt However for all overhanging slopes t he tangent at the Til discharge point must be vertical A summary of t he possible combinations th is assembled in Figu re 6 dll

F SEEPAGE THR~ t GH [HoIOGE~EO tS ISOTROPlcE-RTH D HIS in l mi

a Approximate Solution o~ a ltjO~ The first appro(l mQte matheshy of matical solu tion for determining the quan tity of seepage and the linl Qf seepage through a homogeneous earth section on an impervious base was

tn

L Cqi91lt (J1)

Au~t tltll 9 kymiddot~

Rrl vllJ IJ bull 1f~ ~ (8) For 04 4shyeo 01~ M u-cJ i lrlUonoblr occ~y I CI bull 90middot ~or ong u eIl ~middot d 13 ~rici~ 10 _ 4 ~ on -fViOll (IJ I

CrrP icol ~ufl oF ~ ( l

Flc 1- GIUPUICAL DrnRlII1fAT10~ 0 DISCBIIU POINT TOll a lt60 I)

d proposed independe1 tly in 1918 by Schaffernak (7) in Austria anJ Iterson 1 (8) in Holla nd It is based on Dupuits (9) assumption- that in every point tmiddot of a vertical line the hydraulic gradie~Li~s~nt and e usTie the-sloPe t

h ~~ of the line of sc-eps-ge at its intersection wi th that vertical line This p

e~s u mption r-e pr~ents l good approximation for the average hyuraulic gradien t in such 3 vertical line providing the slope oj t he line of seepage is relatively fllt

C SA C RA-I DE

of a With this assump tio n a nd the condilion that lliLQuan t ity of water uch ng ~h olamph any crosssection per I niLof ti~ must be constan t one 11cal can deri ve the diffe rential equ3tion for t he li ne of 8~pege Irom Figure 78 -iunl dy

q = ky dx (7) the the TbLSOIttioll oCeggajJQLu L yields the_equacio_n_of l parabola Assu ming

ions tha t the qua n tities h d and a in Fi~re 7 are known ~nd ~th tbe bounshydary cond itions y=h for zd and dyjdx =Co n a fo r r= a cos a or y=a sin a irlligra t ion JeaclstQLitoUoVi Qg-form ula [or- t he_di~lance a which detershymlleLthe d ischarge point C of the line of seeQage em the downs tream face

l t hEshy of the dam H of d V ([I h r a= - - - - - - shy (8)

cos Cl wr a sm Q

tq = k a S1n a tan a (9)

T hese equations d iffer from their original form in the use of the disshytance 0 instead of its vertical projection a chailge which provi des a common basis for a ll theorelica l developments in t his paper A fur the r advan tag~

I

(rso n

point slope

This

faulic age is

(owltd

FIG 8 - M Tllo D o r LocTI NG POI-T3 O~ A PAR ABOLA

of t bisect ~~L06e resides in the ~ibili ty of dltp rmi ning g r1piC l ll y_th~

distance a by mellflS of a s im pJ cSlnstppoundti0~ whir Q is sh0 JllfpoundUrure l b T he ordi nate t hrough the k no wn poin t B of the li ne of see page is ex tended to its in tersect ion 1 with the discharge slope a nd a semi~ ircle dra wn thrrlugh the points 1 a nd A ith its cenler on t he discharge s lope Then a horizontal li ne th rough B is inlerec t d li th the discharge s lope in poin t 2 and the d is tance ~-A projected onto the circle yield ing poin t 3 The finll I st ep is to p roject th e d istance 1-] onto the discharge slope T his yields the lesi reu discharge poin t C The pmof for the validity of thi met hod is readi ly Cound by comparing this cons tnIc ti on wit h equation 8) and need no t be disc~S3edin-deta i C- - --shy

lIfJiA cCNtruct ion ie 1i mf1ti5ca ti on cl IDoU1 U mltb c~ - bi ch ~ pr~J in Rdennou 10 ampnd t 1

0 Iy $c

0 e

~ r

J06 S EEPAGE T H ROt GH DHIS

f

FIG 9 - COWPHISON 1lTWZUI B~(c P A R1I 0LA

J

Ii I

~tc1tCn cf lir~ cl~~~=e

307 CA5AG Rmiddot Nll E

v

O 0 - 90 oz ~ - O or a ~JY

44

I I

d~

I I

vshy

rO --ashy

j

tqwton cf

in~ tf S~tPJe

d) AmiddotH) FLOW -ITJ 1OR VRlOlS DISCHIRG S ~O P ~

J08 SEEPGE TH ROUG H DA~S

For thl olution of many problems it is suffi(ien t to know the discharge point of the line of ~Etpnge If it is desired to dr~wJ he en tire li ne of see p g~ f rom the nown- poin t 8 to the di5charge poi nt C one can make use of the

ra h icalme t h~d-h~-in-F i gre 8 fo r the ra-pid-construc tiono f any number of poin ts on a parabola for wh ich a re known two points the tangent to the parabola a t one of thee points and the di rection of the Axis

Through point 8 Figure 8 on l draws a li ne parallll to the axis and determines its int( ~(ction T with the tangent Then one divides the distances B~T nnd C~T in to an arbitrary number of equal parts such as fo ur parts Poi nt Ill and III ilre then connec ted with point C and through points I i nnd J one draws lines paralll l to thl axis T hl points where the linls through 1 and I Q an d II Itc in te rsect are points of t he parabola

b Ap proximate S olullon [or agt ](f The approximate solution by means orequation (8 ) or- the C orre~pondi ng graphiCal method shown in Figure 7b gies sat i~ractory resu lts fo r slo pes of a lt30deg For steeplr slopesmiddot the deviation from the correc t tLlueslntre3Ses ra pidly beyond olerable limits 4

The causes for this deia tion become apparent (rom a study of the Aow net for a slo pe or a = 60deg shown in Figu re 9a Onl can see that in t he vicinity of the dicharge point the size of the squares along the e rtical line through the discharge point decreases only slightly towards the base The average hydraulic gradient along this vertical line is larger than the

dll _ S ~ c hydraulic gradient ~ a long the line of seepage by less than 10 per cent

ds Howevcr th t- ine-of 60 which is the true hydraulic gradient (or the li ne of ~eepnge at the d i~ cl3 r ge point is only aboll t one-halCof the tangent ~( 60deg u ~ed according to Dupuit s assumpt ion Hence the seepage can be analystd wit h a ~a t i5fac tory drgree uf accuracy by m13n of the folloing equ a tion

- d q - k y J ~ (10)

d s

This improe ment a propo~ed by Leo COiagrande (10) The difference be tween the use of the tange nt and the sinp of the slope

of the li ne of seepage is best ill ustrated by the followi ng numr rical comshyparison for various angles

SLoplt 10 n 30 0 S77 0500 eo 73l O9Qti 00 co 1000

H enCl19r slopes lt 3Q both O1fthod may be_wmiddotrdlQr_p raC ical purposes

with equa l adnntagltgt For llop gt 30deg tht deviation by using dy beLomes - - - - dr - shy

I I4 Y J _I

- pound cDfl U d -lt I

in tole m

60deg an to 90deg

Gil which i obtain posi tion from eC)

tF

vertical

ential r readily 12 and doES n metho of see p

A originmiddot graphi in Fig the di 8t rai~i

error middot neglie ~

a lt50 distan constr

is fou 1

1 30

slo~

In ot

the d ischarge ne of seepage lkt use of the lction of any the tangent Ie axis the axis and c divides the parts BuCh as point C a nd The points are poin ts of

sr middott ion by hoa lOwn in steeper 1lolJs ond tol erable

study of the 3tgtC that in t h e ica line

the se Th ger than the

n 10 per cent

nt for tIte linl t ht tangf nt of

C pag t Can bf t ile following

(10)

nc t t ht ~ lupe u mt rical (0111shy

t ie-a l p urpoei dy lng - heromEshy

dx

CASCRAfOE

intolerably large hi le the use of dy is ery sa t isfactory for slopes up w ds

60deg and jf deviations of 25 per cent are permi tt ed it may even be used up to 900

that is for a vertical di5charge fa ce Gilboy (12) succeeded in finding an imQl ic~ solu t ion of ~qua ti9 n (10)

which is recommended where grea ter accuracy is required than can be obtained by m ea ns of the graphical solution The errors involved in the posi t ion of the discharge point as obtained by one or t he other method from equ ation (10) Imiddotere in es tigated by O ~ Reyn tj iens (13)

Using the symbols shoWn in Figure 70 a nd assuming th a t in each y

ver tica l the hydraulic gradient is equal to d equation (10) is the di ffershy----------~----~--------~----d3--------------------en tia l equation for the line of see page T he solu t ion of this equation cannot readily be exprcsed by rec t ngu lar coo rdinates t and y (See Re ferenCe 12 and 13) However t he use of 3 and So mensured a long the line of seepage does no t represent any p ractical di fficu lty in t he act ua l application of this m ethod The ~uan ~i ty a which determ ines the discha r e in t for t he Iin~ of seeps e is fou nd by 0 sim ple in tegrat ion ~ -shy

ky qs =- - + constant

2

Boundary s = a Y = a 3in a q = kasint a Conditi ons s = 30 11 = h

~ - bull ---II---- a = 3 0 -V 30 --- (11)

sin a

q=kasilLa (1 2)

- gai n the qua ntiti es employeu in th ese equat ions differ from the origi nal form as p re~e ntedmiddot in Refere nces 10 a nd 11 to permit a simple gra phica l solution T his graphical so lut ion of equa t ion ( ll ) is i lus trH ~d ilLEigyre 7cnd can be (LSi y verified t requi-es first an assum pt ) n- for t he disc barge poin ~ The length (30 -a ) is sim ply taken equa l to t be straight line from B to CI shown aj a dotted lin e in F igure 7c The Iight error which is int roduced when (o- a) is repla ced by a s t raight line hes a negligi ble effect on the posit ions of the discharge po int In fact for slo~ a lt 60deg it is en ire ly tolerab ie to replace the lengt h So by the straight

dis tance from AB = vh + o t hllS elimina ting trial con middottructions The construction is very simila r to tha t ~ho l n in Figure 7b except that point r is fo und by rota t ing distance CIB or AB aro u nd point A

If d eviations u p to 25_~r centa~ ppoundr lllitt ed the sim plifiM v~l~e

So = vhl+tP = A B may be used also (o r slopes up to 900 For a vert ical bull

stope tneformu la for a~1 feci utedc( t he r~ lu w ing ~ i m pl e fo rm

a = vh + cft -d (13 )

In other word for a ve rtical diich rge face the height of the discha rge

bull

310

~

SEEPAGE T HROU GH DAdS

poin t for the line of seepage can be app roximated by the difference between

the distance AB Vh1+cfZ and its horizontal proj ec tion d

c Solution for a Horizonla~ D ischarge Surfau ~= 180ol In 1931 Professor K ozeny (6) published 8 rigorous solution (or the two-dimenshysional problem of ground water flow over a horizontal imperviou~ ~urface which continues at a given poin t in to a ho ri zon tal discharge face as 5hown in Figure 9d Kozenyi theoret cal Qlutio1 yi~ldsJQLthefiollil1fS-aud e9li-poten tial lines two families of 5onfocal parabolas with point it where the impervious and penous sec tions mee_t ~ the focus

The equation for the line of seepage can be convenien tly expngt8~ed in the follo wing fo nn

yl_ Yo1 x=-- - (14 )

2yo

in wlll ch x and yare the coordinates wih t~eJocu as_ origin and If the Ordi nate a t t he fAcus x = O

lf t he line of see age is detennined by th~oii rdinate~d and h of one bull kpound0i P9in t then_ t he focal gistance ao and the ordinate Yo are computed

JroDLthe folJoving equation

a = Yo = (v ltP+hl-d) (15) 2 13

for which a graphical solut ion is recommended (See Figure l1c ) Th e I

quantity Yo is simply p qual to the difference between the dis tance cr-+hl

fro m the gi en point (d h) ttnh~ focus of the parabola a nQ t he 1bscis~ d The fo cal distance ao is equal to one-half the ordinate Yo shy

In additio1o these simple rela_ti~n hipsj t~f advantage to remember that the tangent to the line of seepage a t x= 0 and y = Yo is inclined at 45 --Thequampntity~G~pag peLuniLO lY idthis according t o Kozeny s solution

q= 2klo= kyo (15)

It is indeed fortunate lhat the problem of seepage llrith a hori zonttl discharge face has such a si mple solu tion not only because of the fact that in modern earth dam and levee design horizont-al drai nage blank e t~ in the downstream section are 8$umi ng considerable impor tance but also because this solution permits fairly reliable a nd sim ple estimates for the po~ition 01 t heJinc or seepage for ove rhanging discharzc slopes

d A pprcrrimate SOlllol(oJLJgr Ovaha1JilllJ_ Dljcl argl_ SurfaceJ (90a lta lt180deg) AJthougb the determinat ion of the line of ~page lnd its point of exi t [or an overhlnging discharge face such ILS a rock fill toc is of importance in the design of elrt h dams little atten tion has been paid to t hiJ p roblem Ecperimental resul ts were published by Leo Casagrunde (1 0 and 11) which pennit 11 re3Sonably accura te delennination oC t he line ot seepage Lster in 1933 the author checked the resul ls of these model -

j

_shyI

r

igtetwe-eo

In 193 1 o-dimenshy

1- sulface s shown lines and A where

~14)

lid the

J of one umputed

(15)

If rhe

~ +hl )~cisa d

member Id at 15deg Iozrnys

( 16)

IlfJrizontai bct that

I in th~ ~fJ I l use I pd tion

Surfaces

uKP and its till toe is

1 pa id to u~u gTJnd e

f)f Ille line ~e model

CSAGRufDpound _ J 11

tes ts by mE nns of graphical sohItions of which a few typical exam ples areshysholn in Figure 9 T hese solutions check well wi th the experimental results just referred to and are illus tra ted for one exa mple in Figure 10 Such studies convinced the author that Forchheimers graphical method for the de termin ation of t he flow net can be util ized for the so lution of seepage problems wil h a free sulface The application of the graphical method to such proble ms requ ires considerable skill T his can be acquired only by extensie use of this method Solu tions such as those shown in Figure 9

-~

- z

~ ~

F Ir 10shy MODE TEn os OJrAWA ~h D RO - gt0 ~lTH OnRH gtCISC

LJIC fI R(1 S LOPE

Xote lone or cllpilllry satllrltill n above lint or ~~Pl ~ t - up~r dy~ lin~ AJI~r L Ca~agrandr (10)

required many hour of wOrk M metim(s s(IEral d~y were s ~nt on one C 3S~ - becJll~~ of thE (o l1lpiicutiol15 that thE unknown uppEr boundlry introdu ces into such ifrpltl prohlpmlt

After sufficipnt graphical 50lut io ns to pprmit l rapid dptErmi nation of thE linr of Sfepagf for any slop( 60deg lt05180deg h ri Iw n accumulatrd the authors att(ntion a r etl ltd to Ko zr n)$ (6) th rorrtical olution for a = 180deg T his proEd a pl ncli ltl opportunit y (or rhe rking tilp accuracy of a purel y graphical olution of it srrp3gc pro lJ r lll with a frN wa ter surface F~u re 9d re prpp nt tbe _origi nal gr3flhird olutioll The diffprrnee helNJubi~oIlt lOll Slid tIl E t hCQrCli ea oluti iLllOlJUQ - ha

~~-x

c ~ nt for alii f)luL0Jl lilll of -1 l 1f Tlwrr[o rf no a tt ntp t 3 madc to in ri llci( t hr thronli(11 i()lutioll ill Fi iZ Irl gr Thi rCIa rbLlc accuracy

Jil SEEPAGE THRQO GH 01 bull

of the gra phical method should convin ce critics that the method i1 not II

plaything but has great merit a nd tha t the ti me spen t on acquiring ~ufficient skill in this method is well in vested

c( ~ di

T o si mplify the application of the graph icd solutio n for very ~ t eep and of

overha ngi ng discharge slopeuro~ such a~ are shown in Figures 9 a band c t hese flow nets were co mpared ~-j t h KOZEgt nys t heoret ical solu tion for a b horiwntal discharge face For th e sake of simplicity the li ne of pepage Cor a = 180deg wh ic h is re presented by eq uations (Uj an d (15) and Figure 9d will be re ferred to as the basic porabola

In F igu re 9 the basic parabola is plotted into every case illustrated The basic parabola and t he Bctunlline of lleepage approach each other very quickly and fo r practical pu rposrs may be ~umed to be id entica l for poin ts whose ordin ates h a re less tha n their ho rizo ntal d i s ttl llce~ from the discharge poi nt C By co mp ring th e actual line of seepage for a glven disch arge sl ope yth t he basiC parabola we fin d th at the in te rsection of this

llrabola wi th the discharge fac e is a di~tance lQ abo ve the discharge poin t

of the line of se~age The ratio c = ~ --shy a+ tla

(see Figure 9) gradually deshy-

cre3Ses wi th incre9li ng an gle a The ratio C is equal to 032 for a =60deg for a vertical surface (a = 90deg) it is 0 26 and for a = 1800 the rat io Cis of course equal to-zero -

In ord er to utilize thtse relati onships for de termini ng the line of s epage and the disch argo point for ste ep vertical an d ov erhnngi ng ~lopeurol there has been plotted in rjgure ll_the reI atio nsh i psJgt~t~een the rati o

c~ ~ a+ la

and th e angle a The quan tity a+ ~l jolt foun d by i n te r~e(ting th e

~~ic Q8 raboI_a witht ~ d ischargeslope an opcmti ~n thal c~n ur ptrformed eithe r grop hicall or mllh emat iclllly In both cases one computes or constructs fi rst Yo = V a- +h~ d for th e kno wn or estimated tart ing point bull of the lin e of seepage The graphi cal determinat io n of the intersec t ion Co is usual( y prefe rred since the bas ic parabola is neecipd fo r t he determi nntion or the line of see page The~o ns tru ti o n of th e p~holJ is be~ t pe rformed Ulih mann~r iUu rtted in Fig re 8 Fo r ta ngent CT~~her the tn n poundUt IIt t he vertex QUkp9caqoia _or _le tangelltyn der 45 at I can be used

= 0 and y = Yo i

middot1middot T he points on the cU ryenamprepresenting the relation betweel a and

M - -c = -- in Figure 110 are deri ved from the graphical so l lJ ti on~ _ _O_+_M Note

how close to a s mooth curve thEse p0ints lie T his il ~no t h cr demon~tratio ll or the degre e of acc urocy that (-an be obtained by m ~ans of the gr1phical met hod

Th e qu a ntity c i not only l funct io n or the a ngie a hil t it a lo ariegt sO f1 ewha t wi til the rcb t i Vc po it iOIl of po i nt i nI or _B and Cu (FiglIrl l Id) T he maltimum variatio ns in c ior the li mits tha t wou ld n ~ r mally 1)( ellshy

(

I

~I

1

1

I

I

I

r shy 1

bull bull bull r

I I

I

0

CASAG RAN DE - JIJ

cou ntered in earth damgt a re about plusmn 5 per cent ThE curve in Figure llb was dctermin ed~(o rilrelatively - short dislanct- fro m t be ent rance t o the d ischarge poin ~ of thp line of seepage in consideration of the importance of stra tifi ca tion in earth darns wh ich is discussed in t he next cha pte r

Haing plotted t he basi c parabola and determined the discharge point by meanl) or t e e-cr-reJa tionFi~re I) and knowing the tangn t to the seepage line a t t he discharge poin t it is an easy ma tter to draw vit h a fai r

a egreeopoundaPproXlmatIonthe -en t ire-line o(seepage as shown inthe~ arious cases in F igure9 -shy -shy - -shy-

FOoltl 0 (u COHlIVTl ()11 CatJT IIC T fPOll

G ~middottmiddot~MT4

Imiddot middot ~ [tb r bull I Ishy

l i I~JJ f Oil ~(H~

bull

bull middotI middot (~middot) iQ1ll

FIG

~__~ lGLI__ _40_ __ ~ _ __

l

riG II b

-shy - FiG Ii J

II I

ill~ l

IImiddot iiITiT) OA - 10shy

FIr 11 shy PPLl( 7iC~ OF middotB~S IC P RBOL TO DET EPll XTIO -l Of

DIlt(H flC E PO IST or LI~ Of -~PCE

SEEPGE THROlG H DUtS 314

t cJuusJip I f or cps rea lll S(o lf-Quanlll lJ of Seepage D ue to the del entr nre cond ition (o r the line of seep3ge a nd d ue to the fact that DU2uitL he aSs umpion is not 3 lid-poundo the_up_ t rea~wed ~ of a d~~ he line o~ gr see age devia tes from the )ara bolic sh apel For the usual shape of a dam thl lhereJ~ection _oint wit h a~ho rp~rmiddotnt~in th~ ~rst sec t ion of the thi

- une-oLeepage while [or a ert ir nl Cnt rance face t here i ~ only an increase obt in CUrllture ithou t reerlta1 of direction

For an acc urate seepage analy~is thce deda t ions should be taken into sec con~ideration Refe rri ng to F igure lId it would he neccsary to know ill US

adall cr the posi t ion of one -Point on~ par3 bolic cun e in the vicinity an of the entranee point B L laagrande has chotie n the intersect ion BI of thr ordina te through th e r ntrance point wit h the cont inuation of the para- bolie li ne of Sf r pagc and hmiddot exprf ~rd the correction 11BL=_~ as a fu nc tion of d h and lhp ~ lope of thu nt ranrf f3c( - graphical presentati9n (11) OIl

fa cili tat e t he fi nding of the ntgtcl~lt~ a ry co rrection oll n middot SOOllwhat sinwler is the following approach Instead of lIeleeti ng s n

pain t B for the ~tn r t of the theoretiral line of seepage we choose its in tershy m sect ion 8 2 wi th the upstream water len The correspondin g correc tion ol c is abollt y~o Y of the ~0i ~on~ projec tion m of the ups tIea~ slope-or ho for Qerage condi tio ns l~ = O3m T his is easy to re mem ber and dispenses til wi th the necessi ty for tnb les or grnphs fo r the correc tion T hedeterm0atiqn ne of the li ne of seepage is t hen ca rried ou t wit h poin t B1JlS the s tarting [oi nt in The act ual shaplt of th f tirst pu rtion of the line of Hf page st a rting at

l 1 point B can easily be Ike tched in 0 that it lpproachrs gradulll y the

01 parabolic cu rve as shown in F igure l id n(

T he ~ulntity of sfepa~_qrer lit _of Iength can be_compJ-I tcd either A fron1 (qullions (1 2) or (16) If we ~ubst itu t e in t h~e rquations tQeJno]

CE qua ntiti es they aprear in the followin~ form

q Iey h +cr- - va __ It coa a) sin a ( 17) b

and q = k(vd1 + 1t= -d) (1$gt tl t I--Eouhe reaLm_Jj orit y of Cl(S encoun~red in_ e~rt h d1 m design l bot h wfq~tjQll s gie_practically the sanl~es ul t so thn t the simpler equa tion (18 )

should be used for general pu rposes In other words the q uant ity of s(pal2 10is prac t ic 1Iy inde pendent of the dischargl slope and is equnl to thcquan t i~ _ shyt lhat ~sPQridSiothe ~hn~ic pafabob - Only in t hose Clses in wh ich tlie

ct artillg po in LoUhe lil( of-~(page-i- ( r)-n rlf the di~c hargC face will the t

t idi irereIl Cf hel~n tht two eqll ~t i ons Wl rran t the use elf equltion (1 7) 7 shyFQ~ l~mplJlliYpoundJY ra rcl~~c in UJich _the r~s~n ce of tail Wllt Pf 5

must be con~iJe rtd-nt b e-Jrsig n-t he middotdete ~m i n ltio n of the line of sccragc----shyand of th e qUJn t ity can be pt riormed hy di idin ~ thl dam ho rizo ntally a t 0

ta il wate r level int o an up pi r a nd lu Cr sptmiddotcion T he li ne oi seCPJge i ~

dctrml ined for the UpPN 5ec tio n in tht Slnl( mannrr as if the di vid iog li ne WCrt all imjXryiollS boundary T he serpage through the IO( f s(tiof1i~

s I d

L

I

CAS GRAND E JIS

Ir to the detennined by means of Da rcy s law using the ra t io of the diffe rlncein Dupuits head over the ave rage lengt h of path of percolation as the hydraulic

fgt linc of gradien t The total quanti ty of seepage is the sum of the q uant ities floying )f a dam through t he upper section and the lower scction The resul ts obtained by on of the this rather crude approxi mation agree remarkably well with the values incrc9Se obtained from an accurate graphica l solution

T hose readers who are interested in data showi ng how the results of keD into seepage tesLs agree wi th the compu ted line of seepage and seepage quantity

know ill ls ing the methods described in this chapter should consult References 10 vici ni ty and 11 ion B of

G SEEPAGE THRO UGH ANISOTROPIC S OtL3 the parlshyI fu nction By a combination of the various methods of approach that hae been l ion (11) outlined a nd witb Qroper consideration oLboundary conditiOp~Jl~UU shy

marized in F igure 6 and d i cussed rr9reJ n det~iJ in_~ppeEdix L ltne can ~(llcti llg arri ye at a rel iab~determinatiol of thc line of ~eepage througb een the i ls ~ r mo t complica ted cross~ctio ns _of earth j a ms ~he cross-sec tion m ay

middotpct ion l consi t of port ions th widely diJferen t pe rmeabi lities howe middoter each sloplt ur homogeneous section in i~el ~ is assume[to- tgte isolropic-tla t is-possessing d i~pen(s t he sa me permeability in a ll di rections Unfortu natelylthis~pacticall minatirJQ l~yer the case J~n- a uniform clean sand consisting of grai ns of the usual ing point icregulllf shape when placed in a _glass flume for t he purpose of building t rr ill C It T he grain3u p a model dam section does not produce an isotropic mass lIa hl -orientate themsplves in such a manne r that the coeffic ient of penneability is

nomiddot-uniform in all directions but la rger in a more or less horizontal direction~ Id eithcr A~-consequcnce~the entire fl ow ne t is markedly infi uenced- resulting in e knuIn cOfLiiderable dc via tions from the theoretical flo w net for an isotropic

material

(t7) Only by usi ng a very unifornl san consis ting of sphe rical g rains middot and by making tests -on- a suffi ciClltly ~ are scul c to reduce the_gapillary d is=

( lS) tu rbance can one arrive a t tes t fe IJ ItS that are in good agreem E nt v th theo ry F or this relon most of the tests des c ri bed in References 10 and 11

i~n both were carried out on Ottawa standanl sunu ltio n (I~ )

Soils in their ~tur Ir un cii sturbcd condi t ion are a laY8 anirotro piclr rrr ~( middot in regard to ~rmeability even if they co nvey to the eye the impression of q llgtmiddotq iIY bei ng entirely uniform in cha ract er If signs of s tratificat ion are uibleIh i 1ill th en~heyerme bility in the di rect ion o j 5tra tifica t ion may easily be tenr wi ll t hI ti mes grea ter t hll~ that normal to stat i ficlllio~- F or d isti nc tly stratified( 17) soils th i ratio can be very much lerger than te n J il Iltrr

When oils are a rti ficia lly deposited ~ in the con8truction of a damf Ser f)l)J or d ike stratification dev elop to a grNL ter or less degree Such stratificationlnt allyat hni always been recognizld by en gineers ItS lJeing unde~i rab and for this (rpagr l~

l e reason specia l con truction method have been developed to disturb ormiddotirlillp li ne detroy it T he hyd r3ulic-fill c r ~ duri ng it s construc t iol1 is fre4uentl y rct ion i~ stirred with lo ng rods in orde r to break up strl ti ficl tion as m uc h as possible

316 SE E PA GE THROlGH DA~S

Sh ee p~foot rollN) lrp efpctingt in compnr ti ng parth fill s wit ho ut crea ting dis tinct lttra tifir ll tion However in pitE of such precautiona ry mPMures a certa in amount of stra tificat ion rem3in In additio n it is pmc ~i ( 3lly

impo ible to eliminate co nsiderable va riations in t hp gen eral characte r of the ma terial in the borrow pit especially aria tions in permeability which ill re~ult in sub~tan tial ariat ions in the permeabil ity of the da m from layer to layer These cannot be eliminated by thorough ron ing Evtgtn the m ost ctr~ rully constructed rolled enrth dam~ posse~s a conside rably greater average permeabil it y in a ho rizont al than in a vertical direction T he refore t horough investiga tion of variations in thr charac ter of the borrow pi t materials forms an important part of preliminary studies Taking into considETa tion the unce rtaint ies that are always encountered in dealing with soil depo~its and cannot be complrtely eliminated by the most elaborate inngtstigatio ns it is esse ntial that we should be conservative in the DSSumpshytions on which the dCign of an eJfth dam is based This requires sprcial attention to the possib le degree of anisot ropy in the dam

The questioll Qf seepage through anisotropic soils was inmiddotes t iga ted for the firs t time and f olYed by Sa msio~ (H) in 1930 Fortuna te ly the solution j s_simple an d kndt itself re dily to_praC tical applica tion The Ro net of an an isotropic soi l does not possess the usua l characteris tics of 8 Row net Howel~r it can be reduced by the application of an app ropria te geometric transformation to an ordinary flow net Designating the maxim um and minimum coefficient of permeabilit y fo r an anisotropic soil as kmcr and kno it can be shown mat hematical ly (sec Referenre 11J ) that by tran~shyforming the ent ingt (foss-sect ion in such a manner t hat alLdill~ions in_1b_e_

direction o( kllZ are reduced by the (actor ~ or tha t all dill(n l i on~ bull k cu

in the di rection of k nre incrcascd by thc factor ykUthe problim i$ k

again redu(ed to a ~o lu t ion-or-LlIplaCP equation--l n -ot he r ~o rds l the flo ll net in the t ransformed ~c t ion ha5 the sa me ch a rv~ t pris tic no line~ anrshyequipotent ial li nes a~ previouslY-d i r ussed in-t hi- pupltgt r Among ot hr rs Forch heimers grap hical m ~ t hod a nd nTlapproxi matc mr th d~ ~uggp~ t rdmiddot in th is paper are a pplicable to the tr3nsformpd ~Ec tion Aft ~ r ha lmiddoting found the line of seepage or the en tire Row net in the t ransfOITnfd sect ion it is a sim ple matter to project this chllrllcteri4ic Aow net bac k into the true section in hi ch Auw lill ~ nnd equ ipote(1tallin t wilinol gcner) lly intershysec t at righ t angle It shollid be noted lha t the hytlralil ic gTuciicllt nt Iny point of t he flo w net and the magni ludp 01 ~eepLl gc prr5~ur(s can only be determin ed in the t ilJe section while the dis t ri butiu ll of pore p rr ~ ures and of hydros tatic upliit t an b~ d ~ ried fro m either section

The quan ti ty of seeplIr can be com pu ted fro m the tran- fo rm~d ~ ( ctjoll

on the basis of the coeffic ient of permc3b ili ty k - ltk In ku For proof see HeJ erence 16

i

I I

I

--shy

_

CASAGRA N DE 317

creating Pllsures 1 0act ically ~ raeter of ~ 141

Y which I)

am from ~ ven the ~ y grea ter Q 0 en fofe Itrrow pit 1 qd ng into

0~Q-ling with ~ E~ labor3te n $Sump- I

S S illl Cl --lt

I~ OJS~ 0

~ rO jgated for

pound ~ olutioll

w net of fl ow net 0

To~ pomct ric oJ mur Ild lt ~ zk tlnd ltmar (by transhy

on~inthe sect ~ inrnioo ~ E

~ z0- j)roblem i~ lt It IJ

II) 50 I he fl o To 2lt

line~ ano ~ r=r g 01 her ~

-middotmiddottrd~u e-i i n _ und ion it i3 a

) the trur 31t) jntrr shy

_ Ilt at allY In oniy be ~S llr( lId

0 leo rct iOIl

For proo f

SEEPAGE THROUG H DAM SJ18

Further information on seepage through stra tified soils the transforshymation theory and examples may be found in References 15 16 and 1i

The appli ca tion of t he t ransformlltion method is illus trated by the sim ple example of a rolled earth dam with a rock 611 toe shown in figLE) 2 The dimensions a nd slopes of this dam are such tha t if a sui table soil is used hardly any doubt would be raised regarding it s stability T he rock fill toe seems to rep resent am ple p rovision for safe discharge of seepage water Indeed the line of seepage assuming isotropic soil does fa ll well within the downstream face as shown in Figu re 12a (coefficient of permeabili ty in horizont al direction kA equal Lo coefficient in vertical d irect ion k) How- shyever if this dam is ca rel essly built of various types of soils with idely d ifferent pem leabil it ies a structure may well result that is roany times m ore pervious in the hori zontal direction than in the vertical direction In the example shown in F igure 12 kA = 9k was chosen On the right-hand side a new cross-sectioo of the dam is plotted ill which aU horizontal dimenshy

sions a re red uced by the factor Ilk = ~ Then t he line of seepage is k 3

dctcnnined ill acco rdance vi th the me thods out li ned pre-lously and proshyjec ted back into the true cross section As Can be seen in F igu re 1211 the line of seepage for kA = 9k does intersect the dow nItream race wh ich is an undesirable condi tion that may in the course of time lead to a partial or complete failure of the strudure

H RnuRXS ON TH E DESIGN OF EARTH DAMS ND LEVEES

The question may arise of how to c~onstru ct the d ownst ream portion of a simple roll ed earth da m so +hal the line of seepage i ll rem ain a safe distance insi2etl1e downstrearn- (aceoftbe - structure-when only- smaH qu-tit ies o(coarse material-are -available- - A- simple sol ution is sugges ted in Figure 12c in which e pervious blaoke t below the downstream portion oi t he dam is em ployed to cont rol the posi tion of the li ne of seepage to Rny desired extent Suc h a blarUel should be buil t u p as a graded filler careshyfully deignecl to pre en t erosion of any soi l from the dlTa

Whenever a d~m or levee consis ts e~sentially of a u niform section or relatively i[Jpervious soil eg possessing all average coefficien t of permeashybili ty of less than 1 x 10- em per sec the pervious blanket may well be

extended as far as the centerl ine of the st ructu re as shown in F igu re 13c Such a de-3ign would add much more to the s tability of the entire downst ream portion including the underlying founda tion than could be accom~ pJ ished by a substan t ial As e~i_ng of the dOlls tresm slope A lev~e built in 1M COnlenlional manner ulilh a Jouniram slope of 1 on-5 uould possess lesJ 8taampilily (han a CI ll-ltIJt1 patled -e in ~Lh ich Ihe d01LfI3ream slop~ i3 made al sep a1 1 on pound bId Chicn CO ll la i M c filter blanket of Ute type ShO lCn in Fig1lJf l 3c In the eumple illustrated in Figu re 13c it as assumed that a pervious founda tion s tratu m lies beneath the levee and thal the perm~~

nsforshy17

)y the He 12 used

fill toe Iva ter lin the l ity in Howshy

dely timps

m In -hand limenshy

age- _

d proshya the

is an tial or

orlion a sa re smal

ested lort ion co any careshy

~ io n or rnlrshyell-shyc 13c d ownshy~c ( o m-

e b u l~

1)O$sess

made )IL1l In

tn3t a nrea-

CASAGRAr OE

~ Q( Q

I

~

t Q

~ ~ ~ ~ ~

I

~

I lt)

co 1

Q

-

- --

320 Spound pound PCE T HROU C H DAMS

through bilities of the levee material a nd the u nde rlying founda tion are the same

soil is 11with a ratio of k4k = 3 The t ransform a tion- which was req uired for

improl robta ining the flo w net is iluslrat poundd by anothe r exam ple shown in Figure

roils art13a and b The assumptions are the sa me as in Figure 13c except that the

may no filte r blanket is reduced to a longit udina l d rainage stri p vth frequent

tidal to transverse ou tl ets A tile drain may a l 0 be embedded within the co re of

tudi nalt he longitudinal dra inage filter to increase the capacity of the d rai nage

well~ tl system if t he structure consists of re-Ia th ely pe rvious soils T his type of

pr(s~ur(drainage would be employed where the quantities of suita ble materia l fo r cohstruthe draina~e layer are very limited T o obtain the flow ne t the true sec t ion CEntra i was transformed into a steeper section using as t ransformation factor very pe l v~middot~k = Va T he Row neo t was then obtained by Forchhe imers graphical pi le cutmiddot method by gradual approxi mation Note the equi-distan t hori zonta l lines

Iml intersect ing the li ne of seepage These were plot ted before ~tarting t he wi ll not fl ow net In this example the line of seepage is not identica l ith t he bas ic well I pare bola because the surface of the founda tio n on which the structure bui It h rests Is Dot a Row line However it is conven ient to use the basic pnrnbola can thl1as a general guide fo r the first plot of the flow net Af ter a stisfactory exist t Iisolution is found t he RoJ net is projected back into the true cross sect ion

F igu re 13 b ~o a t tem pt is made in this pape r to d iscuss in deta il the important and

in teresting relatio nships that exist between t he sta bility of eart h qams or For dikes and the seepage th rough and beneath t hem H oweve r it should be het 1d

emph ized t hat the forces exert ed by pt rcolating water upon the middotoil can ith IImiddotj

be very appreciab le Bnd re often n maximum in crit ica l plt) int s T hese rlq uir seepage forces are readi ly deterrn~ned from a well-constructed flow net be at lr a nd can then be combined with gra vity fo rces for the s tllbility ana lysigt entire d

Anyone who has made comparat il e s t ud~s of the seepage forces that m) l1middotff ill exis t in dams and thei r fou nd a tio ns must be impressed by the par9 010unt entire Ii impo rt ance of the design of those fea tures t hat control seepage It is not L (1lt1 surpris ing that on the basis of e IT) pi rical knowledge leve~ h ve been conshy euroc t iUIl structed wi th fl at slopts A levee built in t he con ven tional manner of sandy so il vt h Slllpes of 1 on 2 or 1 on 3 would be an unSDfe structure How~1er

subst n tial fla tten ing of the slo ppounds i3 a verj UAL iy way of inc rcSin its ~r

safety Be idei eve n very flat slo ~es do not necessa ri ly provide su fficient safety against undermining particu larly when a levee res on a stratified pervious foun dation I n view of the large expenditures on levee construction middot1 I

lwhic h the next detad e vill bring inves tment in research in this field would pay rich dividends if new desigrul ror levees were develo ped t hat would not onl y be much sarer than those built in t he pas t but considerably less I l

I

_ Ifexpensive The widespread opinion among eTLJineers that in Iart~ da m and

lroe design 8td ion makes 01 safety nee(i3 10 be ret-ised Many fai lu res of levees are due to undermining e~used b y seepa~e

th rough the fo undat ion lnless drJ inage provisions lS 5hown for exam ple in Figure 13 are provided the ISC51tSt concentrat ion oi fl ow lines both

---

e same ~ed for Figure~

hat the requent co re of

raina ge type of rial for section I fac tor m phical ta l lints middoting e ht bosic tructu re )arabo13 f ctory section

can t -1 Ga m or lo uld be joil ca n

Thee lo w TI lt

lflaysi ha t may rom oun t It is not

middotecn conshyr) f su ndy iowmiddot~r

ISi rt It

u tFcient tra tified truction ld would middoto uld not bly Ieampgt

da m and

seepage

CA onA middotDE Jll

th rough a nd beneath the dam occurs at t he downs tream toe where the soi l is not confined As stated before fla tte ni ng the lopes does not greatly improve th is condition In some cases partic ul a rly when the foundat ion Boi ls are porou5 a nd dist inctly stratified drainage p rovisions wi t hin a leyee may not be sufficien t protection In such ci rcumstances i t may be beneshyfi cial to drill frequent holes into the founda tion beneath the fut ure longishytudinal drain and to 611 these holes middotwith coarse ma terial Such drainage wells hae been em ployed already fo r anot her putpose t he rel ief of upward pressu re on an overflow dam (see Reference 18) P roperly designed and constructed d rainage we Il~ would effectiE~ly destroy any serious flo w conshycent ration at the tot of the structure In other cases pa rticula rly for a very pe rvi ous but re-Iatively t hin fou ndation strat um thE u~e of a sheet~ pile cu t-ofI may be the ideal solut ion

Imp ro vem ents in lene design as su~ested he-rE are of a na ture that wi ll not produce in t eril rence with modern cons truction me t hods D rainage we 11lt longitud inal a nd t ransye rse drain or dra inagp b l ankpt~ must all bt built before const ruct ion of a levee is st art ed Thp building of t he leiee can thell proceed in the same ma nner as if t he d rainage structures did not exist thus permihing full use of large drag li ne a nd to wer machines

I SEEPAGE THROUGH CmfPOSlTE SECTIONS

For the purposE of controlling seepag( a nd utilizing available soils to bes t adan tage it i~ uu nIly ne ce$~ [lry to b uild dams of 5Pvera sections wi th widely different eor fficient of ptrm e~bili t y Si nce it is (om man to r(q ui re thal t he ra t io F ml(~bi i t y bet wee n ne i-gh boring sect ions should be at le~ne to t(n it i rart ly n(gt( elS ry to determin-t he flo w-net for t he en tire dam if a carefu l study i m Jdr oC t he I e a~ t pen ious scctions Howshy(ver in so mc cases i t m ay bE n ree~ a ry to de term inc the position of t hE rnlire li ll t of slppagE I n F igurf 14 i~ reprodueCd a n Exa mple give n bY L CIl3gra nde (12) showing the lillpound of ~e-e pugpound fora combi nat ion of to ~iolb~t h th do wn1ren m pe t ion built of lto il wh ie h is fin t imelt more

l r---------~~--~

I A r

l~----------__--~~--~~~-=-~kl=-~~=~~~~~~~~~ o

F IG 14 - LI~E Of ~ iP G FOR rO lC8 1 ~ED ~pound(TIO

exam ple Th ~ CL1f11Pljt~ ~ul t - il-~ prili l n model fgt xP ri mcn t

les both Alia L Cur]ronJe (1Ij

shy

32Z SEEPGE THROveH DYS

pervious than the central section The line of seepage through ~uch a_ rl c9_mposite sect ion is found by changing the nltsumed P2sition for the poi~ of intersec tion of t he line of seepage wi th the boundary un ti l the quant ities fi~vigtbro ~gb-bo tb ections a re the same After the correct position of

tlie1i1leOfsee page has been determined one cnn also develop any de~ired I ~

portions of t he Bow net (I

r nJ CO~PAR ISOS BETWEEi FORCII HEIMERS GRPH1CL ~IETHOD t I

H YDRWLIC -IO DEL T ESTS Al D T HE ELECTR IC d

AlLOGY IhTHOD t I

The purpose of mode tes t fo r seepage studies can be twofold

(1) determination of the flo w net for a given cross-secti oOl assuming that I

the soi l is isotropic (2) determina tion of the flow net if t he model is buiId t up in such a manner tha t it resembles the prototype as to possible tratishy j

fi cation charnc ter of the soils etc 1

For tbe firs t purpose it is essential t hat the ma~erial used shall conis t [

of grai ns as near~ spherical and B5 nearly of one size as possible In addishy J

tion the models must be large enough so tha t the heigh t of capillary rise i ll not di tort t he line of seepage parti cularly in tha t portion oC the flo w ne t near the discharge poin t of the li ne of seepage (See Figure 10) [se of Ottawa S tanda rd Sand hss gi ven good resul ts Since one should not go much below the size of Ottawa S tamb rd Sand on eccount of t he distorting effect of capillary rise one is ob liged to use lLrt ific ial spheres such a~ gl3SS sp he res of appropria te si zes for models con tnin ing sections it h di fferen t coeffic ients of permelbility For such coarse ma terials the ali di ty of Darcy s law m llst b-e checked 11 e rcsalts of careful mode l tes t co nducted wi th su poundh-21~terilJls agree wellwith t he solu tions -ob tained by th e graph ical mtlQQd --1b~_e l~c tric analogy method or r igorojlgt t heore ti ~LsectOlu tio n3 so lar as t he latter nre avai lable

For the second purPOse the testing of rnod~ l simi l1r t the proto type one hi to know fi rst of ~Il ho w the coeffi cien t of ~rm a bili ty vari es in the p otot~ not only in its va rious s~ctions bu t partc d lrly vi th in Elcn Sfct io ll due to aniwtro py Then one must buil l th ~ model to imitate on ~ small scale t hee conditio ns It is a wIS te of t ime and money to build a model usi ng the same so il as in t he proto type vi thout atten tion to the an isotropic conditions in the prototype Such a model noes not represent the p otltl type nor are the results compfrlb le to t he conditions for isoshytropi t mate ri a ls because the inev itab le irregu la ri t i c~ lnd s trati 5ca tion due to the m ethod of building th ~ model are reRected in the result ing flow net to such an ext ent th nt the Ro nct looks very muc h like a beshyginners lt tem pt at em ploying the graphicd methl)d T he reu lu of such mod el teti will li e somewhere between the condition for an isotropic model anel the actual conditions in the prototyP and wi ll tell pract ica ll y no thing that can be of a~istance in ou r problem on the contrnrj such

313

r (

such a e point l n tities ilion of de5ired

OJ

worold Ig that build stratishy

~on

t ad ~ shy

ry rise Ie flow rse of 10t go or t ing ~ gl[ Terenr arcy s I with phical n~ so

ty pe ill the

each on a lild J

I) t hI

r i ~oshy

1 due flow

l beshysuch ropi c

ldly such

CASAG RAN DE

result are often ve ry confusing particularly when the tests are made by men inexperienced in theoret ical a nd gra phical solutions and therefore are unable to in terpre t properly the test results

The elect ric analogy method when used by an experi enced opera tor is a u5eful and accu ra te me thod for direct ck1enniTUltion of fl ow nets belween (JiLmiddoten boundanmiddote~ Composite sections consisting of soils with clifferent permeability can also be inves tiga ted by this met jlOd Unfortunately the method does not permit the dire cl determina tion of the line of seepage for those problems in which the upper surface is no t a fixed boundary Another disadvantage of this method is tha t it requ ireg an accurate apparatusmiddot and the const ruction of a special lesUng model (or every problem If compared shywi th graphical solu t ions the electrical method is more expensive and requires more t ime It should also be mentioned that 0 satisfac tory presenshytation of the results obtained by means of the electrical method requires knowledge and applicotion of the graphical method In some instances the amoun t of work required to transform the tei t resu lts into a good-looking flow net would hs e been enough to produce this flow net by the graphical method without assis ta nce of the electrical apparatus T he graphical method will serve as an excellent check on the electrical method and should be used whenever accurate solutions are sought

Anot her im poAant advan tage of the graphica l method is tha t t he proce~3 of finding t he flow net for a proposed section almos t inevi tably suggeltts changes in the design which would improve t he s tab ili ty of the struc ture and often its economy With some experience in the use of the gra ph icalmiddot method the effects of changes in one pr the other detail of the design can quic kly be ap pmised wi thout the necessity of fi nding the comshyplete flow net for a number of different cross-sections T hus there can be bull explored in Ii short time many poss ibilit ies wh ichwo middotlrJ requi re montl~5 of work with any of the other methods Su ch studics ha c alrcldy indicated desirable changes fro m the conventional design of eur~ - u ~n some or which were brie fly discussed in the precedi ng cha pter

Finally there should be men t ioned the jXclagoglca value of the graphical method It gradually de velops a fee lin g or ins t inct for streamline Row which not only improns in tu rn t h Opeed l nd accuracy with which ful net can be det ermine d bu t also de velo ps a much better un ers tanding of the hydrom ~c h 1 n i l~ of F J ~C _J d ground water movemenl T he inshyves tigato r who is t rained only in the lIie of mechanical methods for anJlp ing seepage problems can chec k his tt on ly by performing addishytiona l tests H e is rarely a ble to detect inatcuracies by the a ppearance of the tet resu lts In contrast to t his the author h3S been able to point out even minor inaccuracies in the re5ults obta ined from model tests as a result of the ltenst for 51 ream line fl ow deC loped by applying t he graphical met hod for YP l rs

In concl uding this d iscus ion th au thor middoti5 hes to emph3li ze the almost obv ious point which neve rt heless is frequently overook lt- d that the

SEEP~ OE TH ROU GH DAMSJ24

investigator should con ider carefully before starting any modpl tests what information hp desires to obtain from these tests In nine casps out of ten he will then come to the conclusion t hat he could obtain the results wi thout tests Particularly in t hose cases where he attempts to evaluate the effect of variat ions in the coefficient of pe rmeability he will arrive at a better conception of t he-probable limi ts within which the seepagp condishytions in the pro totype may vary by making Q carcful-study of t he possible variations in the coe fficient of pe rmeability (eg from studies of the va riashyt ions in the borrow pit ma terial) and then applying these values in graphical solutions utili zi ng the transformation method A model test would yield only one r esult the relation of which to the prototype is often unknown Such a test would certainly not permit a concl usion in regard to the probab le limits thin which the actual flow conditions will vary

The practical application of the graphical method would be promoted if for all typica l conditio ns encou ntered in dam design carefully conshystructed fl aw nets were publ ished The beginner in the use of the graphical method in particular would be greatly assisted and encouraged in his efforts to acquire skill in the use of thi valuable tool

A p PENDIX I

(a) D efle ction oj Flow Lines Due 10 ChaTUJ~inPermtab ilil lj FlolI li nes a re deRected a t the boundary between isotropic soils of diffe rent permeabil ity in such a man ner that the quantity q flo wi ng be tween two neighbouring flo w li nes is the same on bo th sides of the boundury Refe rring to Figure 5 in which the flo w net is plo tted on the bas is of squares for tbe ma teria l 00

the lef t of the bou ndary a nd designat ing by ~h tbe drop in head between a ny two neighbouring equipo tential lines the following relationhip can be set up

4h 4h ~a = k(I - = ~-

- a b

or kl C - =- (19 ) kl b

(I ca b = - - and - - = -shy

sin a si n f1 cas a co middot~ f1 Bgt combining these rela tionship3 one arrives at

c ta n 3 kl - = -- = - (20)b lalla k

Expressed in wo rds the deflectio n of the Ro w lines occurs such that tbe t~ngent of the in terJ~c tin g logics with the boundary is inversely proporshytional to the coefficien ts of prmcllbility Furthermore the 5Quare~ on one side of the boundary change on the ot her side into recta ngles ith the ratio of their mdes equal to the ratio of he coefficients of ~rmeabi l i t y such

dtl tests r~(s out

bullhe results ) pvaluate lmve at a 1ge condishyIe possible the variashygraphical

ould yield unknown rei to the

pre ted fu lly conshygrap hicJI

ied in his

Iin tlt are meJbility gthbourill g F igure 5 l t erial on

between n hip can

(1 9)

(20)

thJt the Iy proporshy-es on one Iith the il it y such

CASAGRAND JZS

that the flo w Challllcl3 are wider in the matcriJ with the smaller coefficient of penneabili ty

It is prob ble that Forrhheimer was the fi rst one to use these relationshyships H owever he never took the t rouble to publish them In 1917 be communicated those relationships to Terzaghi who made extensi e use of them in his founda tion investiga tions of dams and a lso taught them in his course in Soil Mechan ics a t the Massachuse tts Institute of Technology during 1925-29

(b) Tra nsJer Conditions for Line oj Seepage at Boundaries General Remarks L Casagrande (IO) made ue of t he general properties of a flow

Ftc 15- D I RIYTln s 0 Dl~rH Rr E CODIT lO ~

ITO On RH-lt rISG ~LO II

net to allalyzr th ron dit ion at the pnt mnce an d di~(h arg( poi nts of the li ne of ~eep3gr Folloilg the sam g~ lI era l app roach the author detershyminrd thE trllnltff r r o nditi O ll ~ fo r othe r Calt ( lt inrluding thl trnnlaquotr a t the -shybou nd ary bftween -o il of diffNfn t ptgtrmeabil it y The re ult ~ are B$~embled in Figurf 6

To acquain t thC fa r1r r with thl method u~d it wi ll be sufficien t to pr(sent in the followillg fhr cifTiatioll for two typica l rli E

(c) Dschn r)~ InllJ (i ()r(fhnnain g S lo pe In FilI ure J5 is shon the Row nrt ill thr im mrdilt( i rinity of thE di-ltrh Hgr point suffir itnt ly rllbrsd so th at Ro I jll r ~ lllt Njll i[lOt r ll tia l li lliS a pl )(l r st raight The lil ope of I he Jillt ur rrpag( 1 t t i l l di r h r~e point the did 1J rge gradient

- - -=1 i f

Cmiddot shy

r I

i middot

J26 SEE PGE THROliGH tgtA~S

is ass umed a rbi trarily then the fl ow ntt is plotted starting wit h a series of equidistant horizontal lines wh ich represent the head of consecutive equipoten t ial li nes One can see imm edia tely that the assumed discharge gradie nt in Figu re 15 cannot be correct because it is impossible to d ra w squares in t he lower portion of the flow net By setti ng up t he condition that the sides a and b of the resulting rectanglts must become equal one can arrive at the necessary condition for the discharge grad ient

By projecting t he sides Il a nd b in the shaded triangles Figure 15 one arrives at the follo wi ng equation

_ b_ sin Y =a cos (ex +( -90deg) = M coila

To fulfill the condi tion a = b the on ly possi ble solution is a = 90- ) that means the lin e of seepage m ust have a ve rtic a l discharge s lope

(d ) Transfer Conditions for Line of Seepage al Boundary bellLeen Soils of Differen t Permeabil ity T o analyze the tra nsfer condi tions fo r the cases illust rated in Figures 6k and m v e stlrt from the conditions that the hydraul ic gradient a t a ny poin along the li ne of seepage is equal to the s ine of the slope of t he li ne of seepage of t hat poin t a nd th~t the quantity shyflo wing t hrough a very t hin flow channel along the line of seepage must be eq ual on bot h sides of the boundary Referring to Figure 16a we have t he folloing velocities along the line of seepage on both sides but in the immeshydiate vicinity of the boundHY

t r bull

VI = 11 sin (a- wi) Vz k1 sin (tJ -Wi )

The quantity uq flowing throughthe channel is

i1q = ak l sin (a-wi) = ck sin (f3-w )

wherein the quantiti es a a nd c rppresen t the wid ths of t he flow channels in accordance with Figure 1630 After repl acing t he quant it ies a and c by their p~oject ion onto the boundaryand substit u ting kdkl = tan~tan a one arri ves at the general condition shy

CO S a sin (d - Wi)

cos 3 si n (a - u) or

sin (gO- a) sin (f3- w ) sln (90 - 8) sin (a -wi)

_ L I~ l ~ 0 ) Hence the only possi ble solu tion is ~ l l QM- rKbull

gO -a = tJ-w or J = 90+w -a (21)or t3 = 2iOo- ll -w

n es ive rge

caw ion one

one

)oila ases

the I the ltity t b the -shy

lmeshy

gtls in c by

1 one

(21)

327

Since th is conditionidoes not contain the coefficipnts of permeability it has to be fulfil led simultaneously with equation (20) Equatiolli (20) and (21) determine Cor a given slope w of the boundary the un known angles a and f3 between the line of seepage and the boundary

(a)

c

(b)

o~~----~-r~shy

jJ = 90 + U)-c(

FlO 16shy TILSSPEA CO~DIT10S 01 LIS 01 SIlUGJ

AT OVElUIA~GI~O BOUNI1RT

The solution of these two equatio ns cm best be found Paphicaily in the manner iIl u5trated in FigHe 16b A ci rcl~ is d ra-vn ogt th an arb itrary radius and t he ang le ADD i3 made equa l to (ltO +w) Then lines a re drawn through poin t3 A and D perpendicul r to the corresponding ~dii T he

-

SEE PGE T H ROUGH DA)[S JIS

problem is to dral another line th rough the cen ter 0 (shown as dot-dash line) which fulfi lls the condition that th p ra tio ABCD = kdk 1bull Such a line can be found q uickly by trinl T he unknolnl ang les a and (3 are de termined by the angles between the dot-dash li ne and li nes 0 4 and OD respectively

Dependini on whether kl is larger or smaller than kt we arrive at solutions in which either point B or point C is nearer the cen ter of the circle The corresponding deflection of the line of seepage is illus trated in Figures 6k and m

Vhi le this theoret ical solution fo r kgtkl can easily be verified by model experiments it is not generally t rue for kl ltt In this case when the downshystream section is more perous the boundary condition for the line of seepage is also influ enced by all other dimensions of the dam especially by the eleva tion of the discharge point and its distance from the boundary under considerAtion Only in special CQ$IS part icu llrly for high tail water

a) h)

FIG 17- TRSrtR CO~DITIOf 01 LI~t or Stpound PGE

AT OmiddotER H ~O I~G B OlThDRf-level and coe fficien ts of permeability that do not differ grttt ly does the Iine or stepage follow the theoret ical solu tion Whenever t he theoret ical solution has th~ appear1nce shown in Figu re 17a with the line of Sefpage deflected into an overhanging slope it re presents a condition t hat may be observed on a small scale in t he laboratory but does not occur on a large scale Instead of the con t inuous line of seepage of Figure I ) a disconshytinuity de middotelov lIth the water seeping ve rt ically into the more perviou1 oil and only incompletely fil li ng its voids In othe r words the quantity dischargi ng vert ically downward at the boundary is inufficient to fill the voids of the coarse m aterial Therefo re normal atmospheric pressure will ct along that sect ion of the boundary and the laws for open dilcherge are valid for ing the line of seepage to assume a ve rtical discharge gructien t at the boundary T hat port ion of t he coarser soil which is only part ially satu rated is illustrated in Figu re)7b by the shadetl area

T he grnphical olut ion hown in Figure 16b also pe rmi ts determination of the transfe r conditions for the en trlnce of the line of seepage for the speurocil case il lustrated in Figure 6c T he opeuron body of wa ter on the upshy

5t re

equ tha i dln

gTq

(

as dot-dash Such e line

determined respecti vely middote arrive a t of the circle d in Figur~

ied by model ~n the downshy

~ the line of - pecialy by 1e boundary h tail water

middot tlgtmiddot does thEgt c t heo ret ical

1110 of seepage that may he

u r on a Io rge i1 ~ disconshy~O L ~~ rv i ou)

the quantity _Ill to fill the pressure will

d isC horge are Hg-e graoient ltmly lIti~ lly

letc rm in tio n middotpge far the

middot~ r on the up-

CASAG RSDpound

329

st ream side may b( cansidlred a parous ma terial with kl shy CIJ for wh ich CD~e FigUftgt ~16b yields J = 90 The $anlEgt conclusion may be reached from equation (21) rc memberingtnat for th is CIlstgt cr = w This reSult means t ha t the line of seEgtpage enters perpentiicu l3rly to the ups tream rac~or the dam as shown in F igure 6c

(e) Singular Points in Q Flow Set In tryin to apply Farch h(imer graph ical mlthod the beginner is freque nt ly puu lld by th E fact t hat ltorne

6

8

FIG 18 - [ u t ~ rR TlOS~ 0 [S] t lR P O[ --rs Of fLO - ~

squares have no rtse mbilnce to rea l squares and that in ~orne Cnes shyflow ti nes nnd eqli polential li nes Cl no t int erect at ri ~ht an~ l es For eXlm ple in Figur( ISa til pound full-dra wn a rea 1 23 4 and 45 6 7 dQ not appear li ke q ua re - to the inexpe riencEd Hop -er by ubdid ci ng sllch arra hy rq ltJlllllOlw rs or uxil itry Rol linE and Eq uipotentiul lin ps one ca n r u- ily r lHck whrthrr the u r i ~ i n al arpl i ~ ~ flur as defi npd ior Ao n Et~ By ~u (h ~llh-di middot i io ll Oil mll~ t Jrri E at lr(~~ whi r h 3ppelr mall and more like rptl qlltit HU II Enr in ma t ra~ it j u ffidElIt to rompr~

r- t~middot r

1

SEEPAGE THROU GH DAMS 330

the average distances betw~n opposite sides that is eg the length 9-10 and 11- 12 by means of a pair of di viders

In F igure 18a the entire Ilrea to the right of points 2 4 7 mut also be considered a square in spi te of the facts t hat the fourth poin t lies a t infinity and that the angle between the flow line and equipo tential line at thie poi nt is zero instead of 90deg H is indeed possible to con tinue subdiidin g this area as sho n by the dot ted lines al ways leaving a sem i-infinite ~ t rip as the last square By this process oC ubdivision t he amoun t of water entering into t he last squa re is cont inuollsly red uced and approaches zero In this way it is pos~ib l e to reconcile the irregularity of the fourth corner by the fact that there is no flow of wate r at thut po int

Simila r irregularities in t he hape of squares a ppelr wherever a given boundary of the soil with wa ter entering or d ischarging and boundary flow lines (im pe rvious base or line of seepage) intersect at a prede termined angle If this AIJ gle is less than 90deg t he n t he velocity of the water at the poin t of in tersection is zero Such point are t he entrance poin t A of the line of seepage in Figu re 18b and points Band C in Figure ISc On t he other hand if the intersect ing angle is greater than 900

t hen the theoretical velocity in that point is infinite Such pOin ts are co rner 4 in F igure 9d co rne rs Band C in Figure 18b and point D in F igure 18c The I~t represhysenti ng the concentration of now lines nt the eleva tion of tail wn ter level is t he cause for the well-known erosion wh ich is observed on the downstream slope of ho mogeneous dam sections at the line of wett ing

At points where the theoretical veloc ity is infin ite the actual Hlocity is influenced by the f cts that for larger veloci ties Da rcy s law lo~s its validity and tha t changes in velocity hld become so im portRnt that the cannot be neglected Hence in the vicini ty of such poin ts the genernl differen tial equa tion (4) is not valid and t he flo w ne t will dcv ipte fro m the theoretical shape However the are35 aITec tc ~c ~0 n1~1l that thc~ p deviations may be d isregarded

ApPE nD II

AddilioTl3 to the Origi1Ull Pa per

(a) Graph ical Procedure for Determining I nlersecl io n between Drscha~Je Slope and Basic P arabola The intersect io n be tween the di~churge face and the bas ic parabola d~s ignafeCl ln Figure 9 by point Co can be d~te rmin ed by the following simple graphical proccdu re

T he ord inate hi of the intersection of t he basic paraboh I = Y= - Jo

21)0

with the discharge s lope J = == I Ian C is (ound li the solu tio n of t hee two equations in the following form

hi = plusmn ~ +Vpound + )1 ta n a tanJa~

~~~

+ p

r

(

19ths 9-10

ust a13o bt a t infinity

t his pain t iclin g this lte totr ip as t o f water

lpproaches tht fourth

oer a gi veil boundary

je tErrnined atel the It A of t he middotc On the theoretical F igure 9d last represhyater level 0 [ am

lal eloc i ty ~ 10e5 its t th3t they the gen eral re from the thOlt th e~ e

- shyf Di ch ar -E fac e and determined

y~-y I= - - shy

2yo 1i th Ee two

C-ISAGRND E JJI

The fi rt member ~ is equal to the distance EB = j in Figure 19 the ~na

second member under the-squsre root is equa l to the distance AB = g the ordinate hi of the intersection is sim ply equal to the sum (J+g) for angle a lt90deg and equal to the difference (g- f) for angles agt 90deg T hese relashytionships are expressed by the construction shown in Figure 19 which need~ no further explanation

T he discharge poillt of the line or seepage is then roun~ as discussed in Sect ion F-d wit h the hel p of Figure II

(b) Com parison bellum Hamdl Theoretical S olution Ilnd the P roposed ApprOIimate M elhods H amel (19) has succeeded in arriving a t a rigorous mathematical solut ion of the problem of seepage through l homogeneous

I r- D shy

FOR c gt 90rOR cr lt90

FIG 19- G R~rHIcL M ETfOD f OR DET ER)[IG I TE RSECTIO~ 8 ampTWEI f

BAjlC P AR80L Agt O Dl~CH RG amp FC~

dam 5ection C nfortunat ely the theory is so cumbe~umE that in its present form it is or li ttle use to the engineering p ro fE5-~io n It will be neel ary to co mp ute a sufficien t num ber of typica l C ~t an d publish the

4

res lI lts in the form of tables or gr~phs before enginNCS wi ll be able to realizE ~ the adva ntages of this theo ret icli t rel tment Recently a few et-( ha e been computed by M uskal (20) for coffer J am sec tions middotith ertical side These solutions presented a n opport un ity to inves tig te at least for 3 few special c ~ts t hE (lccumcy of the ap proxi ma te methoJs proposed in this paper T he resul ts of this com parison were so encouraging that they art presented in the fo llowing paragraphs to perm it the reader to Fo rmulate hi own (o ndu~i o n~

In Figure 20 are asImbled thrEe or the six cas- wh ich were publishM by Mu~kat (20) In eac h C2se th plenltion of t he diot hHgE point a$ comshyputed from H lmrl theory is dlS igl1ted by Co e nd its vertiral distmce

4J shy

I ~

332 lEEP~GE THROUGH D I5

from the tail wa ter level or from the im pe nious base in the absence of tail water is designa ted GR

An approximate elevat ion of the discharge poin ~ was found by means of the graphical procedure shown in Figure 7c To faci litate the comparison betwee n these fig ures all poilltl j n F igu re 20 are marked to corres pond to t hoe in Figu re 7c The contructi on is shown wit h fullli neS7 Th~resultant discharge point is marked C and its eleva lion fro m the base or the tail watlf level is marked a

1-1___ d ---_ - -shy-

Fl O 20shy CO )J P lISO- ltETW Elt RIfOROCS S O A p PROXlldnz

DETE~IlATlOSS OF D IS CH ARGE P OI S T

In addition to this -constmc tio n t he simpli fied p rocedure was used in wh ich 3 0 ft =b i middot For 8 verti c d ischlrgp face the si m plified formulgt for a beco mes a = h+a- - rl as pro pos d by (1eny (6) The correshysponding const ruction is shown in Figure ~IJ by dash lines a nd the resulting cii charge point a nd ele 8t io n are de ignated by C (nd a respect ively

T he case illustrated in Figure Wl corresro ndi ng to Muska t s case L 0 6 is iden ti fied b y the ratio d h = 0937 Hamels theo ry yields the quan t iti es oHh = 0 394 and for the rate of SEgtePage qH = 0539 kh

As was shown by ~lusb t (20) and D acher (23) the rate of seepage ft 1

1 -h J 1 bull bull

com puted by means of Dlpuits form u q = k or WIthout t211 2r1

hl k II F hwater q = 2d represents an eltce nt approxmatlOn or t e c~~e ill usshy

trteJ in Figure 201 we have

q = kh = 0539 kh 2Ii

CAgtA GIUNDpound JJJ

The next C3iC Figure 20b r orrepo nds to ~[ ubt ~ casc 0 and is iuentificd by the rat io d h = 0556 The theory by Hamel yields the quantities aHlh = 0596 and qll = 0898 kh T he approxi ma te rate of seepage comp uted from Dupuit s fo rmula a d e1cri~d fo r the prp ioult ca e is q = 0900 kh

The t hi rd case shown in Figure 20r is identical with M uskat casi 10 2 It differs from the other exam ples by t he assum p t ion of a definitE tail water level and is identi fied by the quan t it ies dh l = 0 663 and ril h1 = 281 From the theory we get aHhl = 0301 and qH 0717 hi Dupuits fo rmula yieldgt q = 0695 kh l bull

The com parison betwee n the valulS for t he eh~ at ion of t he dicharge poi nt obta ined by H a mels ri goro lls solution and thoSt by t he a pproximate methods shows that for engi neering purposes the app roximate solutions a rC cry lnti$factory It is interest ing and of prnc ica l val ue to note that t he a pproximate n et hodl a lso gi e sat isfactory r ul ts for rat ios dh con- o-i dfrably smaller than 10 Considrring that the u ps tream port ioll of the ftOIY net diffe rs considerabl y f rom Dupuit a-u mp tion of a ron middottant hyd raulic grad ient in all ver tical thi reuIt i ~ somew hat un expected For ra tios of dl h lt lO it appears t ha t Kozenys formu ls (13) giHs slightly het ter results tilan the formul a by L Casagran de

The remarkable agreement between the t he-oreti cal ra te of ee~ag( and Dupuits approxi mate solurimrde~rves special emph~i

(c) Graphical Soiwiof by Jl ea ns of th e H odograph A nrw gmphieal method for determ ining the flow net was propofle d by Veillig a nd Shields (30) in whieh t he flo-net is determined gra ph ically in the hodograph pJan( and t hen projected into the ac tu a l eros sect ion

The hod ogrll ph of a flo w line is th E cure which one obtain when plott ing fro m one origin middote locity ec tors fo r a ll the point s of the fto lin f T herefore thp straight li ne connec t ing the origin wi th one point on thr hodograph represe nts the magnitud~ and direc ti on of the velocit y for the correoponding point on the flo w li nr

Since t hE vclor ity a long the f rel watN su rhce is propo rtio nal to the si ne of the s lope thegt hoci0gm ph for the line of seepage is a circ1c with diamete r equal to th~ cOl ffi cie ll( of pe rmeab ility The hodogr ph for a st raight boun d ary is n ~ t rai ltt li e Therr iore a ll boundaritS of the hodoshyg wph that correspo nd to t l~e fl o rugtt of a ho rno~e neo us iso t ro pic d a m ~ectio n are k nown and it i rf)~i b l e to 5(t u p eq tlntions t hrt re pre~e nt the solution of t he p rublrgt m in implicit form T hat stich a theoret ical solution is ra th er complica ted (en for the ~ i mplrst dam XctiOll has oren mellshyt iollcd uefoTe in the disclI- iu n of J-It ml ls t ll rory (19) Therr fo re Weinig anel Shields follow the throre ti r al approach u ing the hoctograph ~ far as mat hema tics pr rmits cOIl( ni r nt ly thc ll t hey p rorr ed to fi nd the flo lin rs a nu pq uipot etlt ial lin eS ill t he hodngra rh bY l graph ica l procedure whic h i~ e~se n tia ll y si mill r to Forrhhl imers meth od

middot

JJ4 SEEPA GE THRO GH DA~S

One advantage of the method by Weinig a nd Shields is the possibility of determining nu merically correct values for the velocity a t certain poin ts along the bou ndaries I n comparison with Forc hheimers graphical method (I

the approach by Weinig and Shields is much more complicated and requires ~middot89 1a thorough acqulintance with the hodograph which very few engineers middot 1

possess F urthermore this method is limited to simple cross-sect ions while or Mi Forchhei mers method can be appl ied to complicated dam sections and ( fou nda tion conditions grea (

Weinig and Shields (30) hae sohcd l ste p triangullr dam middotsection (

Sickby means of the gra phical solution or the hodogrnph T his cross-5ection i I

koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the

FIG 2l- DISC XRI1E P O[XTS OBT~ I SE O BY GIH PIC L ~-oLUTIO (

Of HODOCtP Il -0 M THOO [ LL lST RTE D IS FIG i

illustra ted in Figure 21 Point C rep resents the d ischarge poin t as det ershyi Im ined from the hodogra ph a nd po in t C th e discharge point ~3 i ng the method

shown in F igure 7b The eleva tion of po int Cis 15 pe r cen t lower than that of C How m uch of this d ifie rencc is due to inaccuracy in one or t ht oth er method is u ncertain Probably the hodogrlph solution is mo re accu ra te

when the entrlnce poi nt of the line of seepage is e ry clOSI~ to the di~ch arge li face I n F igu re 20 as well ~ in Figure 21 t he di s c h lr~ plt)ints obta ined

by the sim p le graphical procedure a re situated lower tha n t he other more accu rate solutions T his would ind ica te t he necessity for applying a com~cshy s lion in those cases where up tre3ffi and downst ream face are very close

Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

(I) Ter lamplh i K- v Der Gru ntlbruch 3ll Sl3urnauern und ~ioe Verhtet uog Die Wass rkrLt lm

T erzalZh i K v Erdbaum~hanik ien lll 1925 Terzllghi K v Effect of ~Iinor GPOlogic Detampila ou the Safe t o[ Dams Am last

of ~ia a nd ~re t a l Engrs Technical P ublica t io n 02 15 Feb 1929 (2) Terzagb i K v Auflrieb und Kamppillanlruck a o betooiert eo T alsperreo 1 Can

gre des Grand Barrages Stockholm 1933 (3) T erugbi K v Beans pruchuog voo Gelli cb tM umauern durch da3 stroemeaue

Sickerwasser Die B utcch nik 1934 o 29 (4) Te rzaghi K y Dcr S panaungszWllaad im Pomiddotre amiddotampSlaquoe r trocknender Betonshy

kocrper D er Bau ingenieur 1934 N o 2930 (5) l~orchbelmer P hilipp HydraulLk t h ird edition 1930 (6 ) K o zeoy J G rund w ~rbeegu n bei freiem Spiegel FlU3S- und Kanalycrshy

Bickerung W lSCrkraft u nd MScrwirtsch ft 1931 lo 3 (i) Sch afIerOJlk F C eber d ie S ta nds ichcrheit d urcblaeMiger geschuclteter Daemme

Allgemeine Baucilung 19 Li (8 ) Iterso a F K Th van Een ige thcorcti5Cbc bec houwiogeo oer kwel De

l ngenie ur 1916 5Jld I 19 (9) Dupuit J Etudes lbo retiques et pra tique sur Ie movement des ea ut Paris

18D3 (10) Casagrande Leo 1~eherun gs metboden zur Bc~timmuog von Art und ~(e n6C

der S ickerung dutch gesch uetle te Dse III me Thesis Tteho icb~ Hochschule Vie nna July 1932

Th is p1l~ r amp3 t raoslated into E nblish for u~p in tb~ U S Corps of Engineers by the ~tn f of t he t S a te rgtays Etperimco L StAtioo Vicksbu rg M ill

(11) CBSar3 nde Leo achlrun gsvenlhren lll r F rmit t lung de t Sic kerung in geschUltte te n Dacmmc n lIu f 1I nUll rc hla eltsigc r ~ ohle Die Baut~chnik 1934 lo 15

(t2) Gilboy Glennon H ydrauli c- Fil l Dams I Congres de Grands Barrages Stockhol m 1933

Gilbo) Glennon M cch(lIIiCJ of Hydraulic-fill Da ms Journal olthe Boston So-c of Civ il E ngro Jul y 10)

(1 3 ) Reyn jicn9 G P lci~ 1 E c~rimcn ts on the Flow of Wate r th rough Pervious Soil Th esis ~15 l nst of Teh n~lopound ohy I ()11 ~-~

([ 1) nm~iof A Frey Ei n f1ult~ on Hohrbrun ncn suf Jie r~w ~~tn g tics G rundshy middotmiddot~Ip

Wasser Zeitsch rirl [Jet 8nl~ ll Il Jtt h thrlll tile und Mecblo l L~Jl o ~~~~~middoti middot (1 5) Dachl ~ r Robltrt C el)cr Sic l(e rlSe -~ otmungen In geschichte tem oh terig l - 1-

Die Wa~e rirtltch ft 1 ~3J I) _ F (1 6 ) ~ h lf1em k Frie ri ch Fors rb ~ g d~r ph Y i li l i5c h ~ n G~tze nach ~elcheo

di nurr h~ic ke run( d W1-~rl durch tine Tiliperr oJ~r dur(h den tr tetiru nd middot 1 suufi ll Jet D ie Jlt~ rA ir tsc hJrt 1l3l 1) 30

(17) CeClIgTJnJe Arthur D i~cUi on f) f E W wne s p ~j~ r on Seunt fcom lndc ecp gt PrQ~~ inir- Art ~c C i Enp ~l3 r~ h 1935

(IS) Ter~hi h y Dscu ion of L F Hs rs pape r on Cplift nod Seepage under Dms on Sand Prot~d i ngl-m c C iv Engrs h n 1931

(1 9) Hamel G leher G ru lld asfr tr~rnung Zei u chrift f lngey- ~btb u M ech Yo 14 ~o 3 19J I

ihmcl C and Gunther E _ umerisrhe D urcbrerh nu ng zu dtr AbhapJlung ueb t GntIlJlIasset3 troem unl Zeit~r hrif t r n3~ Milth u ~Ielh Vol 15 1935

I

JJ6 6 EEPGE T HROUGH D uJS

m pqrtanl P uhl iwli(gtTlJ on Ute QIUJlicm 0 Sapag arul ILl Efftct on lILt Sabl li ly of Soil Uta ilaLv p~aetl S ince Ut iJ Pa~r FaJ W rillw (Si( A p~ndie fl)

(20) M usllti t ~J The Sec~ of Water t hroull h D ams lImiddotith Verticol Fac~ Physic Vol 6 ~ 1935

(21) Wyckoff R D and Reed n W Electriol Conduction M odels for the Solushytioo of W e~r ~page Problems PhysiC Vol 6 Dec 1935

(22 ) Knap peD T T and P hll ippe R R Prac ticU So il ~I echanics II~ M uskingum Engg NeWS-~ord Apri19 1936

(23) Dacbler R G rundwMIerstroemung Vienna HJ36 (24) T erughi Kv Simple T e ts w De~rmine HydrosLltic Vplift E ngg ~esshy

Record June 18 1936 (25) Terugili K C ritical lIcigh t illld Facto r of Sa f ~ ty of Slopes againH Sliding

Proc Int~ Coni on 3ltJ il Mcch and Found Eng Vol I bull 0 Gi (2G) T erzaghi K v Dist rib ution of the La ternl Pressure of Sfl lHl on the Timbering

of Cuts P roe I nt Con on Soil Mech and Found Eng Vol I ~o J-3 (27) V~~n burgh C G J pound1ectric In ves t ig1 t ion of Ind ~rgrowl d Wa ter Flov

Nets Proc I n t Con on Soil ~Ie ch Md Found Eng Vol 1 ~o K -1 (23) Vreedenburgh C G J On the ~teady F low of Wa~r Prr~ ol lt i n g throu ~h SoiLs

ith Homogen~u-A n iso tropic P ermeability Pwc Int Conr ampJ d ~l ecb and Found Eng Vol 1 ro K-2

(29) B rU1 l 7 J H A P ressures title to Pereol ting Wale r and T heir Infl uence upon Slrf$CS in Hydrlulic S trtctures Second Congress on Large Dams Wa1 hington D C

B

Vol

1938 (30) Weini bull F snd Shi eld

stroemung d urch ~ taudaemme

- A Graphisch Verfahren zur E r mi ttlun g der SickershyW sserkrnft und W~rwirts chJt 1936 K o is

pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I

296 SEEPAGE THROCGH O YS

J TR 0 0 U C1 10 N

t n t il about ten years ago the design of ear th da ms and dikes was bosed a lm os t exclusi e ly on empirical kn owled ge a nd consi13 ted largely of adopting the cross-section of successful dams Vti tb little regard to differshyences in charac t er of soil an d fo unda tion conditions At p resent we are in a period in wh ich the behavior of dams particu la rly those which have fai led is a nalysed in the li ght of modem soil m echanics T he understanding and k nowledge thus accumulating is being used as the basis for a more scien t iIie approach to t he des ign of such ear t h structu res

T he most outstanding progress in this subject rela tes to the question of seepage beneath d a ms and dikes and to the effect of see page on the stability of t hese st ructures F oundation fa ilures due to seepage com monly known as piping were for the fi rst t ime correctly expla ined by Terzaghi (1)- i

who denlo ped what m ay be termed t he m echa nics of pip ing Later Terzoghi (2 3 a nd 1) co lled_a t1 ention to the importance of t he forces c reated within earth dams and concre te dams d ue to the- percolation of shywater The p ract ical application of this in form ation h as lagged behind our understanding of thee for ces pa rtly because of t heoretical difficul ties of anal y zing p roblems of seepage I t is on ly in recen t y ean t ha t substantial progress has been m ade in the solution of prob lems of seepage and ground wa t e r flo w wi t h free or open surface of the flow t h rough anisotropic mashyterial s and of the condi tions of fl ow throug h join t pla nes of d ifferent materials

B DARCYS L W FOR THE F LOW OF YATpoundR TH ROUGH S OIls

The flow of water through soils so far as it affects the ques tion of seepa ge th rough dams follows Igtarcys empi rical la w which s ta tes that tile amou nl oLflOtL i~ diredly_proportional to the hydraulic gradient_ ThUi la w can be expres~ed ei t her in ~he form

(10 = k i or (1)

Q= kiAt

in which the sym bols have the fol lowing mem ing

(I = discharge veloci ty t k = coeffi cien t of permeabili ty -shyi = hyd raulic gradient

Q= q uan tity of wa ter A c= ares t = time

In Figure 1 t he memi ng of D arcys la w is illustrated in simple form A p rism atic or cy lind rical soil a m p l is etposlt d on the left sidt to 3 head

middot - umerAlJ reler 10 Ill bibllOCaphy1 b cd oI lW pa per

tThia mud Dot be eonl UM-d th ttw ~P-Cft m rxit)O _ plusmn in - bieh -ntio ~ ~WD 01 Toidl r

14 ToIn 01 oIJd mat ter Thbullbullnun roocitr lhMoucgt tgt toil iI rIutltd by tbe _~ niocity IW the dixhMp nloci ty dCI ~ Ibt Qlla3 titT g( lo

bull



r a more




(1)

Ilc form n head


0 elodty

Corcys Law

for i1w Ihroui SoiI



fl

r


I ll elx ~c11 J Jy

PH~ltr tUM3

i CIjltJ



~ 297












(2)




d dllv= xshydt


ok u= k-ax

ak v= k shy

- oy shyah

w= k shyoz


tio (3)

5 1

amp r bull

Figmiddot

ineu m05t

thee lique f the

ly of

I(h 3shy


Thi dare

the

(2)

I thi3 ty o

JI rfh - shye dy

(3)

CASAGRAlDE


o~h oh oh - +-+ - = 0 (4)ox oy OZ






rf

-C~~ ~ WNIlt null

Q-A ~ ~

417 -n

i l J3



















fro m Darcys Ja 1

(5)




aI =shy

~h (6)

J







--

1

r-

(

I

1

I

bull 1

I

J

middot 1

I I bull

I

)

~

CABAORA~OE J01



















0








71 t



~ c_

~ -shy~ ~ bull 1M

Fin

-


ltlint A


-




between


rent soib

CASAGRUmE



~-~~ I


) r)------- ----Jr ) Jr


_- 0- II-~ I) c ) )

m


SU C l





tn

L Cqi91lt (J1)








C SA C RA-I DE





cos Cl wr a sm Q



I

(rso n

point slope

This

faulic age is

(owltd





0 Iy $c

0 e

~ r


f


J

Ii I



v

O 0 - 90 oz ~ - O or a ~JY

44

I I

d~

I I

vshy

rO --ashy

j

tqwton cf

in~ tf S~tPJe










- d q - k y J ~ (10)

d s






I I4 Y J _I


in tole m

60deg an to 90deg


tF

vertical





is fou 1

1 30

slo~

In ot





n 10 per cent



(10)



dx

CASCRAfOE








2


~ - bull ---II---- a = 3 0 -V 30 --- (11)

sin a

q=kasilLa (1 2)






a = vh + cft -d (13 )


bull

310

~






yl_ Yo1 x=-- - (14 )

2yo




a = Yo = (v ltP+hl-d) (15) 2 13





q= 2klo= kyo (15)



j

_shyI

r

igtetwe-eo

In 193 1 o-dimenshy


~14)

lid the

J of one umputed

(15)

If rhe

~ +hl )~cisa d


( 16)



Surfaces






-~

- z

~ ~


LJIC fI R(1 S LOPE




~~-x





c( ~ di









c~ ~ a+ la



= 0 and y = Yo i





(

I

~I

1

1

I

I

I

r shy 1

bull bull bull r

I I

I

0

CASAG RAN DE - JIJ







l i I~JJ f Oil ~(H~

bull


FIG

~__~ lGLI__ _40_ __ ~ _ __

l

riG II b

-shy - FiG Ii J

II I

ill~ l























s I d

L

I

CAS GRAND E JIS









material
















i

I I

I

--shy

_

CASAGRA N DE 317


Y which I)



S S illl Cl --lt

I~ OJS~ 0

~ rO jgated for

pound ~ olutioll










0 leo rct iOIl

For proo f










nsforshy17

)y the He 12 used


dely timps

m In -hand limenshy

age- _

d proshya the

is an tial or

orlion a sa re smal



e b u l~

1)O$sess

made )IL1l In

tn3t a nrea-

CASAGRAr OE

~ Q( Q

I

~

t Q

~ ~ ~ ~ ~

I

~

I lt)

co 1

Q

-

- --




















I




---





Thee lo w TI lt



ISi rt It


da m and

seepage

CA onA middotDE Jll





l r---------~~--~

I A r

l~----------__--~~--~~~-=-~kl=-~~=~~~~~~~~~ o




shy







AlLOGY IhTHOD t I









313

r (


OJ


~on

t ad ~ shy


ty pe ill the

each on a lild J

I) t hI

r i ~oshy

1 due flow

l beshysuch ropi c

ldly such

CASAG RAN DE









A p PENDIX I



4h 4h ~a = k(I - = ~-

- a b

or kl C - =- (19 ) kl b

(I ca b = - - and - - = -shy


c ta n 3 kl - = -- = - (20)b lalla k


dtl tests r~(s out




ied in his


between n hip can

(1 9)

(20)


CASAGRAND JZS









- - -=1 i f

Cmiddot shy

r I

i middot







t r bull





CO S a sin (d - Wi)





n es ive rge

caw ion one

one

)oila ases


lmeshy

gtls in c by

1 one

(21)

327


(a)

c

(b)

o~~----~-r~shy

jJ = 90 + U)-c(




-





a) h)




5t re

equ tha i dln

gTq

(












CASAG RSDpound

329



6

8



r- t~middot r

1







ApPE nD II




21)0



~~~

+ p

r

(

19ths 9-10








y~-y I= - - shy

2yo 1i th Ee two

C-ISAGRND E JJI





I r- D shy





4



4J shy

I ~




1-1___ d ---_ - -shy-






1 -h J 1 bull bull




q = kh = 0539 kh 2Ii









middot







koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the







Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I



r a more




(1)

Ilc form n head


0 elodty

Corcys Law

for i1w Ihroui SoiI



fl

r


I ll elx ~c11 J Jy

PH~ltr tUM3

i CIjltJ



~ 297












(2)




d dllv= xshydt


ok u= k-ax

ak v= k shy

- oy shyah

w= k shyoz


tio (3)

5 1

amp r bull

Figmiddot

ineu m05t

thee lique f the

ly of

I(h 3shy


Thi dare

the

(2)

I thi3 ty o

JI rfh - shye dy

(3)

CASAGRAlDE


o~h oh oh - +-+ - = 0 (4)ox oy OZ






rf

-C~~ ~ WNIlt null

Q-A ~ ~

417 -n

i l J3



















fro m Darcys Ja 1

(5)




aI =shy

~h (6)

J







--

1

r-

(

I

1

I

bull 1

I

J

middot 1

I I bull

I

)

~

CABAORA~OE J01



















0








71 t



~ c_

~ -shy~ ~ bull 1M

Fin

-


ltlint A


-




between


rent soib

CASAGRUmE



~-~~ I


) r)------- ----Jr ) Jr


_- 0- II-~ I) c ) )

m


SU C l





tn

L Cqi91lt (J1)








C SA C RA-I DE





cos Cl wr a sm Q



I

(rso n

point slope

This

faulic age is

(owltd





0 Iy $c

0 e

~ r


f


J

Ii I



v

O 0 - 90 oz ~ - O or a ~JY

44

I I

d~

I I

vshy

rO --ashy

j

tqwton cf

in~ tf S~tPJe










- d q - k y J ~ (10)

d s






I I4 Y J _I


in tole m

60deg an to 90deg


tF

vertical





is fou 1

1 30

slo~

In ot





n 10 per cent



(10)



dx

CASCRAfOE








2


~ - bull ---II---- a = 3 0 -V 30 --- (11)

sin a

q=kasilLa (1 2)






a = vh + cft -d (13 )


bull

310

~






yl_ Yo1 x=-- - (14 )

2yo




a = Yo = (v ltP+hl-d) (15) 2 13





q= 2klo= kyo (15)



j

_shyI

r

igtetwe-eo

In 193 1 o-dimenshy


~14)

lid the

J of one umputed

(15)

If rhe

~ +hl )~cisa d


( 16)



Surfaces






-~

- z

~ ~


LJIC fI R(1 S LOPE




~~-x





c( ~ di









c~ ~ a+ la



= 0 and y = Yo i





(

I

~I

1

1

I

I

I

r shy 1

bull bull bull r

I I

I

0

CASAG RAN DE - JIJ







l i I~JJ f Oil ~(H~

bull


FIG

~__~ lGLI__ _40_ __ ~ _ __

l

riG II b

-shy - FiG Ii J

II I

ill~ l























s I d

L

I

CAS GRAND E JIS









material
















i

I I

I

--shy

_

CASAGRA N DE 317


Y which I)



S S illl Cl --lt

I~ OJS~ 0

~ rO jgated for

pound ~ olutioll










0 leo rct iOIl

For proo f










nsforshy17

)y the He 12 used


dely timps

m In -hand limenshy

age- _

d proshya the

is an tial or

orlion a sa re smal



e b u l~

1)O$sess

made )IL1l In

tn3t a nrea-

CASAGRAr OE

~ Q( Q

I

~

t Q

~ ~ ~ ~ ~

I

~

I lt)

co 1

Q

-

- --




















I




---





Thee lo w TI lt



ISi rt It


da m and

seepage

CA onA middotDE Jll





l r---------~~--~

I A r

l~----------__--~~--~~~-=-~kl=-~~=~~~~~~~~~ o




shy







AlLOGY IhTHOD t I









313

r (


OJ


~on

t ad ~ shy


ty pe ill the

each on a lild J

I) t hI

r i ~oshy

1 due flow

l beshysuch ropi c

ldly such

CASAG RAN DE









A p PENDIX I



4h 4h ~a = k(I - = ~-

- a b

or kl C - =- (19 ) kl b

(I ca b = - - and - - = -shy


c ta n 3 kl - = -- = - (20)b lalla k


dtl tests r~(s out




ied in his


between n hip can

(1 9)

(20)


CASAGRAND JZS









- - -=1 i f

Cmiddot shy

r I

i middot







t r bull





CO S a sin (d - Wi)





n es ive rge

caw ion one

one

)oila ases


lmeshy

gtls in c by

1 one

(21)

327


(a)

c

(b)

o~~----~-r~shy

jJ = 90 + U)-c(




-





a) h)




5t re

equ tha i dln

gTq

(












CASAG RSDpound

329



6

8



r- t~middot r

1







ApPE nD II




21)0



~~~

+ p

r

(

19ths 9-10








y~-y I= - - shy

2yo 1i th Ee two

C-ISAGRND E JJI





I r- D shy





4



4J shy

I ~




1-1___ d ---_ - -shy-






1 -h J 1 bull bull




q = kh = 0539 kh 2Ii









middot







koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the







Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I












(2)




d dllv= xshydt


ok u= k-ax

ak v= k shy

- oy shyah

w= k shyoz


tio (3)

5 1

amp r bull

Figmiddot

ineu m05t

thee lique f the

ly of

I(h 3shy


Thi dare

the

(2)

I thi3 ty o

JI rfh - shye dy

(3)

CASAGRAlDE


o~h oh oh - +-+ - = 0 (4)ox oy OZ






rf

-C~~ ~ WNIlt null

Q-A ~ ~

417 -n

i l J3



















fro m Darcys Ja 1

(5)




aI =shy

~h (6)

J







--

1

r-

(

I

1

I

bull 1

I

J

middot 1

I I bull

I

)

~

CABAORA~OE J01



















0








71 t



~ c_

~ -shy~ ~ bull 1M

Fin

-


ltlint A


-




between


rent soib

CASAGRUmE



~-~~ I


) r)------- ----Jr ) Jr


_- 0- II-~ I) c ) )

m


SU C l





tn

L Cqi91lt (J1)








C SA C RA-I DE





cos Cl wr a sm Q



I

(rso n

point slope

This

faulic age is

(owltd





0 Iy $c

0 e

~ r


f


J

Ii I



v

O 0 - 90 oz ~ - O or a ~JY

44

I I

d~

I I

vshy

rO --ashy

j

tqwton cf

in~ tf S~tPJe










- d q - k y J ~ (10)

d s






I I4 Y J _I


in tole m

60deg an to 90deg


tF

vertical





is fou 1

1 30

slo~

In ot





n 10 per cent



(10)



dx

CASCRAfOE








2


~ - bull ---II---- a = 3 0 -V 30 --- (11)

sin a

q=kasilLa (1 2)






a = vh + cft -d (13 )


bull

310

~






yl_ Yo1 x=-- - (14 )

2yo




a = Yo = (v ltP+hl-d) (15) 2 13





q= 2klo= kyo (15)



j

_shyI

r

igtetwe-eo

In 193 1 o-dimenshy


~14)

lid the

J of one umputed

(15)

If rhe

~ +hl )~cisa d


( 16)



Surfaces






-~

- z

~ ~


LJIC fI R(1 S LOPE




~~-x





c( ~ di









c~ ~ a+ la



= 0 and y = Yo i





(

I

~I

1

1

I

I

I

r shy 1

bull bull bull r

I I

I

0

CASAG RAN DE - JIJ







l i I~JJ f Oil ~(H~

bull


FIG

~__~ lGLI__ _40_ __ ~ _ __

l

riG II b

-shy - FiG Ii J

II I

ill~ l























s I d

L

I

CAS GRAND E JIS









material
















i

I I

I

--shy

_

CASAGRA N DE 317


Y which I)



S S illl Cl --lt

I~ OJS~ 0

~ rO jgated for

pound ~ olutioll










0 leo rct iOIl

For proo f










nsforshy17

)y the He 12 used


dely timps

m In -hand limenshy

age- _

d proshya the

is an tial or

orlion a sa re smal



e b u l~

1)O$sess

made )IL1l In

tn3t a nrea-

CASAGRAr OE

~ Q( Q

I

~

t Q

~ ~ ~ ~ ~

I

~

I lt)

co 1

Q

-

- --




















I




---





Thee lo w TI lt



ISi rt It


da m and

seepage

CA onA middotDE Jll





l r---------~~--~

I A r

l~----------__--~~--~~~-=-~kl=-~~=~~~~~~~~~ o




shy







AlLOGY IhTHOD t I









313

r (


OJ


~on

t ad ~ shy


ty pe ill the

each on a lild J

I) t hI

r i ~oshy

1 due flow

l beshysuch ropi c

ldly such

CASAG RAN DE









A p PENDIX I



4h 4h ~a = k(I - = ~-

- a b

or kl C - =- (19 ) kl b

(I ca b = - - and - - = -shy


c ta n 3 kl - = -- = - (20)b lalla k


dtl tests r~(s out




ied in his


between n hip can

(1 9)

(20)


CASAGRAND JZS









- - -=1 i f

Cmiddot shy

r I

i middot







t r bull





CO S a sin (d - Wi)





n es ive rge

caw ion one

one

)oila ases


lmeshy

gtls in c by

1 one

(21)

327


(a)

c

(b)

o~~----~-r~shy

jJ = 90 + U)-c(




-





a) h)




5t re

equ tha i dln

gTq

(












CASAG RSDpound

329



6

8



r- t~middot r

1







ApPE nD II




21)0



~~~

+ p

r

(

19ths 9-10








y~-y I= - - shy

2yo 1i th Ee two

C-ISAGRND E JJI





I r- D shy





4



4J shy

I ~




1-1___ d ---_ - -shy-






1 -h J 1 bull bull




q = kh = 0539 kh 2Ii









middot







koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the







Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I

amp r bull

Figmiddot

ineu m05t

thee lique f the

ly of

I(h 3shy


Thi dare

the

(2)

I thi3 ty o

JI rfh - shye dy

(3)

CASAGRAlDE


o~h oh oh - +-+ - = 0 (4)ox oy OZ






rf

-C~~ ~ WNIlt null

Q-A ~ ~

417 -n

i l J3



















fro m Darcys Ja 1

(5)




aI =shy

~h (6)

J







--

1

r-

(

I

1

I

bull 1

I

J

middot 1

I I bull

I

)

~

CABAORA~OE J01



















0








71 t



~ c_

~ -shy~ ~ bull 1M

Fin

-


ltlint A


-




between


rent soib

CASAGRUmE



~-~~ I


) r)------- ----Jr ) Jr


_- 0- II-~ I) c ) )

m


SU C l





tn

L Cqi91lt (J1)








C SA C RA-I DE





cos Cl wr a sm Q



I

(rso n

point slope

This

faulic age is

(owltd





0 Iy $c

0 e

~ r


f


J

Ii I



v

O 0 - 90 oz ~ - O or a ~JY

44

I I

d~

I I

vshy

rO --ashy

j

tqwton cf

in~ tf S~tPJe










- d q - k y J ~ (10)

d s






I I4 Y J _I


in tole m

60deg an to 90deg


tF

vertical





is fou 1

1 30

slo~

In ot





n 10 per cent



(10)



dx

CASCRAfOE








2


~ - bull ---II---- a = 3 0 -V 30 --- (11)

sin a

q=kasilLa (1 2)






a = vh + cft -d (13 )


bull

310

~






yl_ Yo1 x=-- - (14 )

2yo




a = Yo = (v ltP+hl-d) (15) 2 13





q= 2klo= kyo (15)



j

_shyI

r

igtetwe-eo

In 193 1 o-dimenshy


~14)

lid the

J of one umputed

(15)

If rhe

~ +hl )~cisa d


( 16)



Surfaces






-~

- z

~ ~


LJIC fI R(1 S LOPE




~~-x





c( ~ di









c~ ~ a+ la



= 0 and y = Yo i





(

I

~I

1

1

I

I

I

r shy 1

bull bull bull r

I I

I

0

CASAG RAN DE - JIJ







l i I~JJ f Oil ~(H~

bull


FIG

~__~ lGLI__ _40_ __ ~ _ __

l

riG II b

-shy - FiG Ii J

II I

ill~ l























s I d

L

I

CAS GRAND E JIS









material
















i

I I

I

--shy

_

CASAGRA N DE 317


Y which I)



S S illl Cl --lt

I~ OJS~ 0

~ rO jgated for

pound ~ olutioll










0 leo rct iOIl

For proo f










nsforshy17

)y the He 12 used


dely timps

m In -hand limenshy

age- _

d proshya the

is an tial or

orlion a sa re smal



e b u l~

1)O$sess

made )IL1l In

tn3t a nrea-

CASAGRAr OE

~ Q( Q

I

~

t Q

~ ~ ~ ~ ~

I

~

I lt)

co 1

Q

-

- --




















I




---





Thee lo w TI lt



ISi rt It


da m and

seepage

CA onA middotDE Jll





l r---------~~--~

I A r

l~----------__--~~--~~~-=-~kl=-~~=~~~~~~~~~ o




shy







AlLOGY IhTHOD t I









313

r (


OJ


~on

t ad ~ shy


ty pe ill the

each on a lild J

I) t hI

r i ~oshy

1 due flow

l beshysuch ropi c

ldly such

CASAG RAN DE









A p PENDIX I



4h 4h ~a = k(I - = ~-

- a b

or kl C - =- (19 ) kl b

(I ca b = - - and - - = -shy


c ta n 3 kl - = -- = - (20)b lalla k


dtl tests r~(s out




ied in his


between n hip can

(1 9)

(20)


CASAGRAND JZS









- - -=1 i f

Cmiddot shy

r I

i middot







t r bull





CO S a sin (d - Wi)





n es ive rge

caw ion one

one

)oila ases


lmeshy

gtls in c by

1 one

(21)

327


(a)

c

(b)

o~~----~-r~shy

jJ = 90 + U)-c(




-





a) h)




5t re

equ tha i dln

gTq

(












CASAG RSDpound

329



6

8



r- t~middot r

1







ApPE nD II




21)0



~~~

+ p

r

(

19ths 9-10








y~-y I= - - shy

2yo 1i th Ee two

C-ISAGRND E JJI





I r- D shy





4



4J shy

I ~




1-1___ d ---_ - -shy-






1 -h J 1 bull bull




q = kh = 0539 kh 2Ii









middot







koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the







Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I
















fro m Darcys Ja 1

(5)




aI =shy

~h (6)

J







--

1

r-

(

I

1

I

bull 1

I

J

middot 1

I I bull

I

)

~

CABAORA~OE J01



















0








71 t



~ c_

~ -shy~ ~ bull 1M

Fin

-


ltlint A


-




between


rent soib

CASAGRUmE



~-~~ I


) r)------- ----Jr ) Jr


_- 0- II-~ I) c ) )

m


SU C l





tn

L Cqi91lt (J1)








C SA C RA-I DE





cos Cl wr a sm Q



I

(rso n

point slope

This

faulic age is

(owltd





0 Iy $c

0 e

~ r


f


J

Ii I



v

O 0 - 90 oz ~ - O or a ~JY

44

I I

d~

I I

vshy

rO --ashy

j

tqwton cf

in~ tf S~tPJe










- d q - k y J ~ (10)

d s






I I4 Y J _I


in tole m

60deg an to 90deg


tF

vertical





is fou 1

1 30

slo~

In ot





n 10 per cent



(10)



dx

CASCRAfOE








2


~ - bull ---II---- a = 3 0 -V 30 --- (11)

sin a

q=kasilLa (1 2)






a = vh + cft -d (13 )


bull

310

~






yl_ Yo1 x=-- - (14 )

2yo




a = Yo = (v ltP+hl-d) (15) 2 13





q= 2klo= kyo (15)



j

_shyI

r

igtetwe-eo

In 193 1 o-dimenshy


~14)

lid the

J of one umputed

(15)

If rhe

~ +hl )~cisa d


( 16)



Surfaces






-~

- z

~ ~


LJIC fI R(1 S LOPE




~~-x





c( ~ di









c~ ~ a+ la



= 0 and y = Yo i





(

I

~I

1

1

I

I

I

r shy 1

bull bull bull r

I I

I

0

CASAG RAN DE - JIJ







l i I~JJ f Oil ~(H~

bull


FIG

~__~ lGLI__ _40_ __ ~ _ __

l

riG II b

-shy - FiG Ii J

II I

ill~ l























s I d

L

I

CAS GRAND E JIS









material
















i

I I

I

--shy

_

CASAGRA N DE 317


Y which I)



S S illl Cl --lt

I~ OJS~ 0

~ rO jgated for

pound ~ olutioll










0 leo rct iOIl

For proo f










nsforshy17

)y the He 12 used


dely timps

m In -hand limenshy

age- _

d proshya the

is an tial or

orlion a sa re smal



e b u l~

1)O$sess

made )IL1l In

tn3t a nrea-

CASAGRAr OE

~ Q( Q

I

~

t Q

~ ~ ~ ~ ~

I

~

I lt)

co 1

Q

-

- --




















I




---





Thee lo w TI lt



ISi rt It


da m and

seepage

CA onA middotDE Jll





l r---------~~--~

I A r

l~----------__--~~--~~~-=-~kl=-~~=~~~~~~~~~ o




shy







AlLOGY IhTHOD t I









313

r (


OJ


~on

t ad ~ shy


ty pe ill the

each on a lild J

I) t hI

r i ~oshy

1 due flow

l beshysuch ropi c

ldly such

CASAG RAN DE









A p PENDIX I



4h 4h ~a = k(I - = ~-

- a b

or kl C - =- (19 ) kl b

(I ca b = - - and - - = -shy


c ta n 3 kl - = -- = - (20)b lalla k


dtl tests r~(s out




ied in his


between n hip can

(1 9)

(20)


CASAGRAND JZS









- - -=1 i f

Cmiddot shy

r I

i middot







t r bull





CO S a sin (d - Wi)





n es ive rge

caw ion one

one

)oila ases


lmeshy

gtls in c by

1 one

(21)

327


(a)

c

(b)

o~~----~-r~shy

jJ = 90 + U)-c(




-





a) h)




5t re

equ tha i dln

gTq

(












CASAG RSDpound

329



6

8



r- t~middot r

1







ApPE nD II




21)0



~~~

+ p

r

(

19ths 9-10








y~-y I= - - shy

2yo 1i th Ee two

C-ISAGRND E JJI





I r- D shy





4



4J shy

I ~




1-1___ d ---_ - -shy-






1 -h J 1 bull bull




q = kh = 0539 kh 2Ii









middot







koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the







Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I

--

1

r-

(

I

1

I

bull 1

I

J

middot 1

I I bull

I

)

~

CABAORA~OE J01



















0








71 t



~ c_

~ -shy~ ~ bull 1M

Fin

-


ltlint A


-




between


rent soib

CASAGRUmE



~-~~ I


) r)------- ----Jr ) Jr


_- 0- II-~ I) c ) )

m


SU C l





tn

L Cqi91lt (J1)








C SA C RA-I DE





cos Cl wr a sm Q



I

(rso n

point slope

This

faulic age is

(owltd





0 Iy $c

0 e

~ r


f


J

Ii I



v

O 0 - 90 oz ~ - O or a ~JY

44

I I

d~

I I

vshy

rO --ashy

j

tqwton cf

in~ tf S~tPJe










- d q - k y J ~ (10)

d s






I I4 Y J _I


in tole m

60deg an to 90deg


tF

vertical





is fou 1

1 30

slo~

In ot





n 10 per cent



(10)



dx

CASCRAfOE








2


~ - bull ---II---- a = 3 0 -V 30 --- (11)

sin a

q=kasilLa (1 2)






a = vh + cft -d (13 )


bull

310

~






yl_ Yo1 x=-- - (14 )

2yo




a = Yo = (v ltP+hl-d) (15) 2 13





q= 2klo= kyo (15)



j

_shyI

r

igtetwe-eo

In 193 1 o-dimenshy


~14)

lid the

J of one umputed

(15)

If rhe

~ +hl )~cisa d


( 16)



Surfaces






-~

- z

~ ~


LJIC fI R(1 S LOPE




~~-x





c( ~ di









c~ ~ a+ la



= 0 and y = Yo i





(

I

~I

1

1

I

I

I

r shy 1

bull bull bull r

I I

I

0

CASAG RAN DE - JIJ







l i I~JJ f Oil ~(H~

bull


FIG

~__~ lGLI__ _40_ __ ~ _ __

l

riG II b

-shy - FiG Ii J

II I

ill~ l























s I d

L

I

CAS GRAND E JIS









material
















i

I I

I

--shy

_

CASAGRA N DE 317


Y which I)



S S illl Cl --lt

I~ OJS~ 0

~ rO jgated for

pound ~ olutioll










0 leo rct iOIl

For proo f










nsforshy17

)y the He 12 used


dely timps

m In -hand limenshy

age- _

d proshya the

is an tial or

orlion a sa re smal



e b u l~

1)O$sess

made )IL1l In

tn3t a nrea-

CASAGRAr OE

~ Q( Q

I

~

t Q

~ ~ ~ ~ ~

I

~

I lt)

co 1

Q

-

- --




















I




---





Thee lo w TI lt



ISi rt It


da m and

seepage

CA onA middotDE Jll





l r---------~~--~

I A r

l~----------__--~~--~~~-=-~kl=-~~=~~~~~~~~~ o




shy







AlLOGY IhTHOD t I









313

r (


OJ


~on

t ad ~ shy


ty pe ill the

each on a lild J

I) t hI

r i ~oshy

1 due flow

l beshysuch ropi c

ldly such

CASAG RAN DE









A p PENDIX I



4h 4h ~a = k(I - = ~-

- a b

or kl C - =- (19 ) kl b

(I ca b = - - and - - = -shy


c ta n 3 kl - = -- = - (20)b lalla k


dtl tests r~(s out




ied in his


between n hip can

(1 9)

(20)


CASAGRAND JZS









- - -=1 i f

Cmiddot shy

r I

i middot







t r bull





CO S a sin (d - Wi)





n es ive rge

caw ion one

one

)oila ases


lmeshy

gtls in c by

1 one

(21)

327


(a)

c

(b)

o~~----~-r~shy

jJ = 90 + U)-c(




-





a) h)




5t re

equ tha i dln

gTq

(












CASAG RSDpound

329



6

8



r- t~middot r

1







ApPE nD II




21)0



~~~

+ p

r

(

19ths 9-10








y~-y I= - - shy

2yo 1i th Ee two

C-ISAGRND E JJI





I r- D shy





4



4J shy

I ~




1-1___ d ---_ - -shy-






1 -h J 1 bull bull




q = kh = 0539 kh 2Ii









middot







koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the







Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I





0








71 t



~ c_

~ -shy~ ~ bull 1M

Fin

-


ltlint A


-




between


rent soib

CASAGRUmE



~-~~ I


) r)------- ----Jr ) Jr


_- 0- II-~ I) c ) )

m


SU C l





tn

L Cqi91lt (J1)








C SA C RA-I DE





cos Cl wr a sm Q



I

(rso n

point slope

This

faulic age is

(owltd





0 Iy $c

0 e

~ r


f


J

Ii I



v

O 0 - 90 oz ~ - O or a ~JY

44

I I

d~

I I

vshy

rO --ashy

j

tqwton cf

in~ tf S~tPJe










- d q - k y J ~ (10)

d s






I I4 Y J _I


in tole m

60deg an to 90deg


tF

vertical





is fou 1

1 30

slo~

In ot





n 10 per cent



(10)



dx

CASCRAfOE








2


~ - bull ---II---- a = 3 0 -V 30 --- (11)

sin a

q=kasilLa (1 2)






a = vh + cft -d (13 )


bull

310

~






yl_ Yo1 x=-- - (14 )

2yo




a = Yo = (v ltP+hl-d) (15) 2 13





q= 2klo= kyo (15)



j

_shyI

r

igtetwe-eo

In 193 1 o-dimenshy


~14)

lid the

J of one umputed

(15)

If rhe

~ +hl )~cisa d


( 16)



Surfaces






-~

- z

~ ~


LJIC fI R(1 S LOPE




~~-x





c( ~ di









c~ ~ a+ la



= 0 and y = Yo i





(

I

~I

1

1

I

I

I

r shy 1

bull bull bull r

I I

I

0

CASAG RAN DE - JIJ







l i I~JJ f Oil ~(H~

bull


FIG

~__~ lGLI__ _40_ __ ~ _ __

l

riG II b

-shy - FiG Ii J

II I

ill~ l























s I d

L

I

CAS GRAND E JIS









material
















i

I I

I

--shy

_

CASAGRA N DE 317


Y which I)



S S illl Cl --lt

I~ OJS~ 0

~ rO jgated for

pound ~ olutioll










0 leo rct iOIl

For proo f










nsforshy17

)y the He 12 used


dely timps

m In -hand limenshy

age- _

d proshya the

is an tial or

orlion a sa re smal



e b u l~

1)O$sess

made )IL1l In

tn3t a nrea-

CASAGRAr OE

~ Q( Q

I

~

t Q

~ ~ ~ ~ ~

I

~

I lt)

co 1

Q

-

- --




















I




---





Thee lo w TI lt



ISi rt It


da m and

seepage

CA onA middotDE Jll





l r---------~~--~

I A r

l~----------__--~~--~~~-=-~kl=-~~=~~~~~~~~~ o




shy







AlLOGY IhTHOD t I









313

r (


OJ


~on

t ad ~ shy


ty pe ill the

each on a lild J

I) t hI

r i ~oshy

1 due flow

l beshysuch ropi c

ldly such

CASAG RAN DE









A p PENDIX I



4h 4h ~a = k(I - = ~-

- a b

or kl C - =- (19 ) kl b

(I ca b = - - and - - = -shy


c ta n 3 kl - = -- = - (20)b lalla k


dtl tests r~(s out




ied in his


between n hip can

(1 9)

(20)


CASAGRAND JZS









- - -=1 i f

Cmiddot shy

r I

i middot







t r bull





CO S a sin (d - Wi)





n es ive rge

caw ion one

one

)oila ases


lmeshy

gtls in c by

1 one

(21)

327


(a)

c

(b)

o~~----~-r~shy

jJ = 90 + U)-c(




-





a) h)




5t re

equ tha i dln

gTq

(












CASAG RSDpound

329



6

8



r- t~middot r

1







ApPE nD II




21)0



~~~

+ p

r

(

19ths 9-10








y~-y I= - - shy

2yo 1i th Ee two

C-ISAGRND E JJI





I r- D shy





4



4J shy

I ~




1-1___ d ---_ - -shy-






1 -h J 1 bull bull




q = kh = 0539 kh 2Ii









middot







koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the







Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I

-


ltlint A


-




between


rent soib

CASAGRUmE



~-~~ I


) r)------- ----Jr ) Jr


_- 0- II-~ I) c ) )

m


SU C l





tn

L Cqi91lt (J1)








C SA C RA-I DE





cos Cl wr a sm Q



I

(rso n

point slope

This

faulic age is

(owltd





0 Iy $c

0 e

~ r


f


J

Ii I



v

O 0 - 90 oz ~ - O or a ~JY

44

I I

d~

I I

vshy

rO --ashy

j

tqwton cf

in~ tf S~tPJe










- d q - k y J ~ (10)

d s






I I4 Y J _I


in tole m

60deg an to 90deg


tF

vertical





is fou 1

1 30

slo~

In ot





n 10 per cent



(10)



dx

CASCRAfOE








2


~ - bull ---II---- a = 3 0 -V 30 --- (11)

sin a

q=kasilLa (1 2)






a = vh + cft -d (13 )


bull

310

~






yl_ Yo1 x=-- - (14 )

2yo




a = Yo = (v ltP+hl-d) (15) 2 13





q= 2klo= kyo (15)



j

_shyI

r

igtetwe-eo

In 193 1 o-dimenshy


~14)

lid the

J of one umputed

(15)

If rhe

~ +hl )~cisa d


( 16)



Surfaces






-~

- z

~ ~


LJIC fI R(1 S LOPE




~~-x





c( ~ di









c~ ~ a+ la



= 0 and y = Yo i





(

I

~I

1

1

I

I

I

r shy 1

bull bull bull r

I I

I

0

CASAG RAN DE - JIJ







l i I~JJ f Oil ~(H~

bull


FIG

~__~ lGLI__ _40_ __ ~ _ __

l

riG II b

-shy - FiG Ii J

II I

ill~ l























s I d

L

I

CAS GRAND E JIS









material
















i

I I

I

--shy

_

CASAGRA N DE 317


Y which I)



S S illl Cl --lt

I~ OJS~ 0

~ rO jgated for

pound ~ olutioll










0 leo rct iOIl

For proo f










nsforshy17

)y the He 12 used


dely timps

m In -hand limenshy

age- _

d proshya the

is an tial or

orlion a sa re smal



e b u l~

1)O$sess

made )IL1l In

tn3t a nrea-

CASAGRAr OE

~ Q( Q

I

~

t Q

~ ~ ~ ~ ~

I

~

I lt)

co 1

Q

-

- --




















I




---





Thee lo w TI lt



ISi rt It


da m and

seepage

CA onA middotDE Jll





l r---------~~--~

I A r

l~----------__--~~--~~~-=-~kl=-~~=~~~~~~~~~ o




shy







AlLOGY IhTHOD t I









313

r (


OJ


~on

t ad ~ shy


ty pe ill the

each on a lild J

I) t hI

r i ~oshy

1 due flow

l beshysuch ropi c

ldly such

CASAG RAN DE









A p PENDIX I



4h 4h ~a = k(I - = ~-

- a b

or kl C - =- (19 ) kl b

(I ca b = - - and - - = -shy


c ta n 3 kl - = -- = - (20)b lalla k


dtl tests r~(s out




ied in his


between n hip can

(1 9)

(20)


CASAGRAND JZS









- - -=1 i f

Cmiddot shy

r I

i middot







t r bull





CO S a sin (d - Wi)





n es ive rge

caw ion one

one

)oila ases


lmeshy

gtls in c by

1 one

(21)

327


(a)

c

(b)

o~~----~-r~shy

jJ = 90 + U)-c(




-





a) h)




5t re

equ tha i dln

gTq

(












CASAG RSDpound

329



6

8



r- t~middot r

1







ApPE nD II




21)0



~~~

+ p

r

(

19ths 9-10








y~-y I= - - shy

2yo 1i th Ee two

C-ISAGRND E JJI





I r- D shy





4



4J shy

I ~




1-1___ d ---_ - -shy-






1 -h J 1 bull bull




q = kh = 0539 kh 2Ii









middot







koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the







Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I





tn

L Cqi91lt (J1)








C SA C RA-I DE





cos Cl wr a sm Q



I

(rso n

point slope

This

faulic age is

(owltd





0 Iy $c

0 e

~ r


f


J

Ii I



v

O 0 - 90 oz ~ - O or a ~JY

44

I I

d~

I I

vshy

rO --ashy

j

tqwton cf

in~ tf S~tPJe










- d q - k y J ~ (10)

d s






I I4 Y J _I


in tole m

60deg an to 90deg


tF

vertical





is fou 1

1 30

slo~

In ot





n 10 per cent



(10)



dx

CASCRAfOE








2


~ - bull ---II---- a = 3 0 -V 30 --- (11)

sin a

q=kasilLa (1 2)






a = vh + cft -d (13 )


bull

310

~






yl_ Yo1 x=-- - (14 )

2yo




a = Yo = (v ltP+hl-d) (15) 2 13





q= 2klo= kyo (15)



j

_shyI

r

igtetwe-eo

In 193 1 o-dimenshy


~14)

lid the

J of one umputed

(15)

If rhe

~ +hl )~cisa d


( 16)



Surfaces






-~

- z

~ ~


LJIC fI R(1 S LOPE




~~-x





c( ~ di









c~ ~ a+ la



= 0 and y = Yo i





(

I

~I

1

1

I

I

I

r shy 1

bull bull bull r

I I

I

0

CASAG RAN DE - JIJ







l i I~JJ f Oil ~(H~

bull


FIG

~__~ lGLI__ _40_ __ ~ _ __

l

riG II b

-shy - FiG Ii J

II I

ill~ l























s I d

L

I

CAS GRAND E JIS









material
















i

I I

I

--shy

_

CASAGRA N DE 317


Y which I)



S S illl Cl --lt

I~ OJS~ 0

~ rO jgated for

pound ~ olutioll










0 leo rct iOIl

For proo f










nsforshy17

)y the He 12 used


dely timps

m In -hand limenshy

age- _

d proshya the

is an tial or

orlion a sa re smal



e b u l~

1)O$sess

made )IL1l In

tn3t a nrea-

CASAGRAr OE

~ Q( Q

I

~

t Q

~ ~ ~ ~ ~

I

~

I lt)

co 1

Q

-

- --




















I




---





Thee lo w TI lt



ISi rt It


da m and

seepage

CA onA middotDE Jll





l r---------~~--~

I A r

l~----------__--~~--~~~-=-~kl=-~~=~~~~~~~~~ o




shy







AlLOGY IhTHOD t I









313

r (


OJ


~on

t ad ~ shy


ty pe ill the

each on a lild J

I) t hI

r i ~oshy

1 due flow

l beshysuch ropi c

ldly such

CASAG RAN DE









A p PENDIX I



4h 4h ~a = k(I - = ~-

- a b

or kl C - =- (19 ) kl b

(I ca b = - - and - - = -shy


c ta n 3 kl - = -- = - (20)b lalla k


dtl tests r~(s out




ied in his


between n hip can

(1 9)

(20)


CASAGRAND JZS









- - -=1 i f

Cmiddot shy

r I

i middot







t r bull





CO S a sin (d - Wi)





n es ive rge

caw ion one

one

)oila ases


lmeshy

gtls in c by

1 one

(21)

327


(a)

c

(b)

o~~----~-r~shy

jJ = 90 + U)-c(




-





a) h)




5t re

equ tha i dln

gTq

(












CASAG RSDpound

329



6

8



r- t~middot r

1







ApPE nD II




21)0



~~~

+ p

r

(

19ths 9-10








y~-y I= - - shy

2yo 1i th Ee two

C-ISAGRND E JJI





I r- D shy





4



4J shy

I ~




1-1___ d ---_ - -shy-






1 -h J 1 bull bull




q = kh = 0539 kh 2Ii









middot







koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the







Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I

C SA C RA-I DE





cos Cl wr a sm Q



I

(rso n

point slope

This

faulic age is

(owltd





0 Iy $c

0 e

~ r


f


J

Ii I



v

O 0 - 90 oz ~ - O or a ~JY

44

I I

d~

I I

vshy

rO --ashy

j

tqwton cf

in~ tf S~tPJe










- d q - k y J ~ (10)

d s






I I4 Y J _I


in tole m

60deg an to 90deg


tF

vertical





is fou 1

1 30

slo~

In ot





n 10 per cent



(10)



dx

CASCRAfOE








2


~ - bull ---II---- a = 3 0 -V 30 --- (11)

sin a

q=kasilLa (1 2)






a = vh + cft -d (13 )


bull

310

~






yl_ Yo1 x=-- - (14 )

2yo




a = Yo = (v ltP+hl-d) (15) 2 13





q= 2klo= kyo (15)



j

_shyI

r

igtetwe-eo

In 193 1 o-dimenshy


~14)

lid the

J of one umputed

(15)

If rhe

~ +hl )~cisa d


( 16)



Surfaces






-~

- z

~ ~


LJIC fI R(1 S LOPE




~~-x





c( ~ di









c~ ~ a+ la



= 0 and y = Yo i





(

I

~I

1

1

I

I

I

r shy 1

bull bull bull r

I I

I

0

CASAG RAN DE - JIJ







l i I~JJ f Oil ~(H~

bull


FIG

~__~ lGLI__ _40_ __ ~ _ __

l

riG II b

-shy - FiG Ii J

II I

ill~ l























s I d

L

I

CAS GRAND E JIS









material
















i

I I

I

--shy

_

CASAGRA N DE 317


Y which I)



S S illl Cl --lt

I~ OJS~ 0

~ rO jgated for

pound ~ olutioll










0 leo rct iOIl

For proo f










nsforshy17

)y the He 12 used


dely timps

m In -hand limenshy

age- _

d proshya the

is an tial or

orlion a sa re smal



e b u l~

1)O$sess

made )IL1l In

tn3t a nrea-

CASAGRAr OE

~ Q( Q

I

~

t Q

~ ~ ~ ~ ~

I

~

I lt)

co 1

Q

-

- --




















I




---





Thee lo w TI lt



ISi rt It


da m and

seepage

CA onA middotDE Jll





l r---------~~--~

I A r

l~----------__--~~--~~~-=-~kl=-~~=~~~~~~~~~ o




shy







AlLOGY IhTHOD t I









313

r (


OJ


~on

t ad ~ shy


ty pe ill the

each on a lild J

I) t hI

r i ~oshy

1 due flow

l beshysuch ropi c

ldly such

CASAG RAN DE









A p PENDIX I



4h 4h ~a = k(I - = ~-

- a b

or kl C - =- (19 ) kl b

(I ca b = - - and - - = -shy


c ta n 3 kl - = -- = - (20)b lalla k


dtl tests r~(s out




ied in his


between n hip can

(1 9)

(20)


CASAGRAND JZS









- - -=1 i f

Cmiddot shy

r I

i middot







t r bull





CO S a sin (d - Wi)





n es ive rge

caw ion one

one

)oila ases


lmeshy

gtls in c by

1 one

(21)

327


(a)

c

(b)

o~~----~-r~shy

jJ = 90 + U)-c(




-





a) h)




5t re

equ tha i dln

gTq

(












CASAG RSDpound

329



6

8



r- t~middot r

1







ApPE nD II




21)0



~~~

+ p

r

(

19ths 9-10








y~-y I= - - shy

2yo 1i th Ee two

C-ISAGRND E JJI





I r- D shy





4



4J shy

I ~




1-1___ d ---_ - -shy-






1 -h J 1 bull bull




q = kh = 0539 kh 2Ii









middot







koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the







Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I

~ r


f


J

Ii I



v

O 0 - 90 oz ~ - O or a ~JY

44

I I

d~

I I

vshy

rO --ashy

j

tqwton cf

in~ tf S~tPJe










- d q - k y J ~ (10)

d s






I I4 Y J _I


in tole m

60deg an to 90deg


tF

vertical





is fou 1

1 30

slo~

In ot





n 10 per cent



(10)



dx

CASCRAfOE








2


~ - bull ---II---- a = 3 0 -V 30 --- (11)

sin a

q=kasilLa (1 2)






a = vh + cft -d (13 )


bull

310

~






yl_ Yo1 x=-- - (14 )

2yo




a = Yo = (v ltP+hl-d) (15) 2 13





q= 2klo= kyo (15)



j

_shyI

r

igtetwe-eo

In 193 1 o-dimenshy


~14)

lid the

J of one umputed

(15)

If rhe

~ +hl )~cisa d


( 16)



Surfaces






-~

- z

~ ~


LJIC fI R(1 S LOPE




~~-x





c( ~ di









c~ ~ a+ la



= 0 and y = Yo i





(

I

~I

1

1

I

I

I

r shy 1

bull bull bull r

I I

I

0

CASAG RAN DE - JIJ







l i I~JJ f Oil ~(H~

bull


FIG

~__~ lGLI__ _40_ __ ~ _ __

l

riG II b

-shy - FiG Ii J

II I

ill~ l























s I d

L

I

CAS GRAND E JIS









material
















i

I I

I

--shy

_

CASAGRA N DE 317


Y which I)



S S illl Cl --lt

I~ OJS~ 0

~ rO jgated for

pound ~ olutioll










0 leo rct iOIl

For proo f










nsforshy17

)y the He 12 used


dely timps

m In -hand limenshy

age- _

d proshya the

is an tial or

orlion a sa re smal



e b u l~

1)O$sess

made )IL1l In

tn3t a nrea-

CASAGRAr OE

~ Q( Q

I

~

t Q

~ ~ ~ ~ ~

I

~

I lt)

co 1

Q

-

- --




















I




---





Thee lo w TI lt



ISi rt It


da m and

seepage

CA onA middotDE Jll





l r---------~~--~

I A r

l~----------__--~~--~~~-=-~kl=-~~=~~~~~~~~~ o




shy







AlLOGY IhTHOD t I









313

r (


OJ


~on

t ad ~ shy


ty pe ill the

each on a lild J

I) t hI

r i ~oshy

1 due flow

l beshysuch ropi c

ldly such

CASAG RAN DE









A p PENDIX I



4h 4h ~a = k(I - = ~-

- a b

or kl C - =- (19 ) kl b

(I ca b = - - and - - = -shy


c ta n 3 kl - = -- = - (20)b lalla k


dtl tests r~(s out




ied in his


between n hip can

(1 9)

(20)


CASAGRAND JZS









- - -=1 i f

Cmiddot shy

r I

i middot







t r bull





CO S a sin (d - Wi)





n es ive rge

caw ion one

one

)oila ases


lmeshy

gtls in c by

1 one

(21)

327


(a)

c

(b)

o~~----~-r~shy

jJ = 90 + U)-c(




-





a) h)




5t re

equ tha i dln

gTq

(












CASAG RSDpound

329



6

8



r- t~middot r

1







ApPE nD II




21)0



~~~

+ p

r

(

19ths 9-10








y~-y I= - - shy

2yo 1i th Ee two

C-ISAGRND E JJI





I r- D shy





4



4J shy

I ~




1-1___ d ---_ - -shy-






1 -h J 1 bull bull




q = kh = 0539 kh 2Ii









middot







koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the







Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I


v

O 0 - 90 oz ~ - O or a ~JY

44

I I

d~

I I

vshy

rO --ashy

j

tqwton cf

in~ tf S~tPJe










- d q - k y J ~ (10)

d s






I I4 Y J _I


in tole m

60deg an to 90deg


tF

vertical





is fou 1

1 30

slo~

In ot





n 10 per cent



(10)



dx

CASCRAfOE








2


~ - bull ---II---- a = 3 0 -V 30 --- (11)

sin a

q=kasilLa (1 2)






a = vh + cft -d (13 )


bull

310

~






yl_ Yo1 x=-- - (14 )

2yo




a = Yo = (v ltP+hl-d) (15) 2 13





q= 2klo= kyo (15)



j

_shyI

r

igtetwe-eo

In 193 1 o-dimenshy


~14)

lid the

J of one umputed

(15)

If rhe

~ +hl )~cisa d


( 16)



Surfaces






-~

- z

~ ~


LJIC fI R(1 S LOPE




~~-x





c( ~ di









c~ ~ a+ la



= 0 and y = Yo i





(

I

~I

1

1

I

I

I

r shy 1

bull bull bull r

I I

I

0

CASAG RAN DE - JIJ







l i I~JJ f Oil ~(H~

bull


FIG

~__~ lGLI__ _40_ __ ~ _ __

l

riG II b

-shy - FiG Ii J

II I

ill~ l























s I d

L

I

CAS GRAND E JIS









material
















i

I I

I

--shy

_

CASAGRA N DE 317


Y which I)



S S illl Cl --lt

I~ OJS~ 0

~ rO jgated for

pound ~ olutioll










0 leo rct iOIl

For proo f










nsforshy17

)y the He 12 used


dely timps

m In -hand limenshy

age- _

d proshya the

is an tial or

orlion a sa re smal



e b u l~

1)O$sess

made )IL1l In

tn3t a nrea-

CASAGRAr OE

~ Q( Q

I

~

t Q

~ ~ ~ ~ ~

I

~

I lt)

co 1

Q

-

- --




















I




---





Thee lo w TI lt



ISi rt It


da m and

seepage

CA onA middotDE Jll





l r---------~~--~

I A r

l~----------__--~~--~~~-=-~kl=-~~=~~~~~~~~~ o




shy







AlLOGY IhTHOD t I









313

r (


OJ


~on

t ad ~ shy


ty pe ill the

each on a lild J

I) t hI

r i ~oshy

1 due flow

l beshysuch ropi c

ldly such

CASAG RAN DE









A p PENDIX I



4h 4h ~a = k(I - = ~-

- a b

or kl C - =- (19 ) kl b

(I ca b = - - and - - = -shy


c ta n 3 kl - = -- = - (20)b lalla k


dtl tests r~(s out




ied in his


between n hip can

(1 9)

(20)


CASAGRAND JZS









- - -=1 i f

Cmiddot shy

r I

i middot







t r bull





CO S a sin (d - Wi)





n es ive rge

caw ion one

one

)oila ases


lmeshy

gtls in c by

1 one

(21)

327


(a)

c

(b)

o~~----~-r~shy

jJ = 90 + U)-c(




-





a) h)




5t re

equ tha i dln

gTq

(












CASAG RSDpound

329



6

8



r- t~middot r

1







ApPE nD II




21)0



~~~

+ p

r

(

19ths 9-10








y~-y I= - - shy

2yo 1i th Ee two

C-ISAGRND E JJI





I r- D shy





4



4J shy

I ~




1-1___ d ---_ - -shy-






1 -h J 1 bull bull




q = kh = 0539 kh 2Ii









middot







koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the







Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I









- d q - k y J ~ (10)

d s






I I4 Y J _I


in tole m

60deg an to 90deg


tF

vertical





is fou 1

1 30

slo~

In ot





n 10 per cent



(10)



dx

CASCRAfOE








2


~ - bull ---II---- a = 3 0 -V 30 --- (11)

sin a

q=kasilLa (1 2)






a = vh + cft -d (13 )


bull

310

~






yl_ Yo1 x=-- - (14 )

2yo




a = Yo = (v ltP+hl-d) (15) 2 13





q= 2klo= kyo (15)



j

_shyI

r

igtetwe-eo

In 193 1 o-dimenshy


~14)

lid the

J of one umputed

(15)

If rhe

~ +hl )~cisa d


( 16)



Surfaces






-~

- z

~ ~


LJIC fI R(1 S LOPE




~~-x





c( ~ di









c~ ~ a+ la



= 0 and y = Yo i





(

I

~I

1

1

I

I

I

r shy 1

bull bull bull r

I I

I

0

CASAG RAN DE - JIJ







l i I~JJ f Oil ~(H~

bull


FIG

~__~ lGLI__ _40_ __ ~ _ __

l

riG II b

-shy - FiG Ii J

II I

ill~ l























s I d

L

I

CAS GRAND E JIS









material
















i

I I

I

--shy

_

CASAGRA N DE 317


Y which I)



S S illl Cl --lt

I~ OJS~ 0

~ rO jgated for

pound ~ olutioll










0 leo rct iOIl

For proo f










nsforshy17

)y the He 12 used


dely timps

m In -hand limenshy

age- _

d proshya the

is an tial or

orlion a sa re smal



e b u l~

1)O$sess

made )IL1l In

tn3t a nrea-

CASAGRAr OE

~ Q( Q

I

~

t Q

~ ~ ~ ~ ~

I

~

I lt)

co 1

Q

-

- --




















I




---





Thee lo w TI lt



ISi rt It


da m and

seepage

CA onA middotDE Jll





l r---------~~--~

I A r

l~----------__--~~--~~~-=-~kl=-~~=~~~~~~~~~ o




shy







AlLOGY IhTHOD t I









313

r (


OJ


~on

t ad ~ shy


ty pe ill the

each on a lild J

I) t hI

r i ~oshy

1 due flow

l beshysuch ropi c

ldly such

CASAG RAN DE









A p PENDIX I



4h 4h ~a = k(I - = ~-

- a b

or kl C - =- (19 ) kl b

(I ca b = - - and - - = -shy


c ta n 3 kl - = -- = - (20)b lalla k


dtl tests r~(s out




ied in his


between n hip can

(1 9)

(20)


CASAGRAND JZS









- - -=1 i f

Cmiddot shy

r I

i middot







t r bull





CO S a sin (d - Wi)





n es ive rge

caw ion one

one

)oila ases


lmeshy

gtls in c by

1 one

(21)

327


(a)

c

(b)

o~~----~-r~shy

jJ = 90 + U)-c(




-





a) h)




5t re

equ tha i dln

gTq

(












CASAG RSDpound

329



6

8



r- t~middot r

1







ApPE nD II




21)0



~~~

+ p

r

(

19ths 9-10








y~-y I= - - shy

2yo 1i th Ee two

C-ISAGRND E JJI





I r- D shy





4



4J shy

I ~




1-1___ d ---_ - -shy-






1 -h J 1 bull bull




q = kh = 0539 kh 2Ii









middot







koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the







Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I





n 10 per cent



(10)



dx

CASCRAfOE








2


~ - bull ---II---- a = 3 0 -V 30 --- (11)

sin a

q=kasilLa (1 2)






a = vh + cft -d (13 )


bull

310

~






yl_ Yo1 x=-- - (14 )

2yo




a = Yo = (v ltP+hl-d) (15) 2 13





q= 2klo= kyo (15)



j

_shyI

r

igtetwe-eo

In 193 1 o-dimenshy


~14)

lid the

J of one umputed

(15)

If rhe

~ +hl )~cisa d


( 16)



Surfaces






-~

- z

~ ~


LJIC fI R(1 S LOPE




~~-x





c( ~ di









c~ ~ a+ la



= 0 and y = Yo i





(

I

~I

1

1

I

I

I

r shy 1

bull bull bull r

I I

I

0

CASAG RAN DE - JIJ







l i I~JJ f Oil ~(H~

bull


FIG

~__~ lGLI__ _40_ __ ~ _ __

l

riG II b

-shy - FiG Ii J

II I

ill~ l























s I d

L

I

CAS GRAND E JIS









material
















i

I I

I

--shy

_

CASAGRA N DE 317


Y which I)



S S illl Cl --lt

I~ OJS~ 0

~ rO jgated for

pound ~ olutioll










0 leo rct iOIl

For proo f










nsforshy17

)y the He 12 used


dely timps

m In -hand limenshy

age- _

d proshya the

is an tial or

orlion a sa re smal



e b u l~

1)O$sess

made )IL1l In

tn3t a nrea-

CASAGRAr OE

~ Q( Q

I

~

t Q

~ ~ ~ ~ ~

I

~

I lt)

co 1

Q

-

- --




















I




---





Thee lo w TI lt



ISi rt It


da m and

seepage

CA onA middotDE Jll





l r---------~~--~

I A r

l~----------__--~~--~~~-=-~kl=-~~=~~~~~~~~~ o




shy







AlLOGY IhTHOD t I









313

r (


OJ


~on

t ad ~ shy


ty pe ill the

each on a lild J

I) t hI

r i ~oshy

1 due flow

l beshysuch ropi c

ldly such

CASAG RAN DE









A p PENDIX I



4h 4h ~a = k(I - = ~-

- a b

or kl C - =- (19 ) kl b

(I ca b = - - and - - = -shy


c ta n 3 kl - = -- = - (20)b lalla k


dtl tests r~(s out




ied in his


between n hip can

(1 9)

(20)


CASAGRAND JZS









- - -=1 i f

Cmiddot shy

r I

i middot







t r bull





CO S a sin (d - Wi)





n es ive rge

caw ion one

one

)oila ases


lmeshy

gtls in c by

1 one

(21)

327


(a)

c

(b)

o~~----~-r~shy

jJ = 90 + U)-c(




-





a) h)




5t re

equ tha i dln

gTq

(












CASAG RSDpound

329



6

8



r- t~middot r

1







ApPE nD II




21)0



~~~

+ p

r

(

19ths 9-10








y~-y I= - - shy

2yo 1i th Ee two

C-ISAGRND E JJI





I r- D shy





4



4J shy

I ~




1-1___ d ---_ - -shy-






1 -h J 1 bull bull




q = kh = 0539 kh 2Ii









middot







koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the







Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I

310

~






yl_ Yo1 x=-- - (14 )

2yo




a = Yo = (v ltP+hl-d) (15) 2 13





q= 2klo= kyo (15)



j

_shyI

r

igtetwe-eo

In 193 1 o-dimenshy


~14)

lid the

J of one umputed

(15)

If rhe

~ +hl )~cisa d


( 16)



Surfaces






-~

- z

~ ~


LJIC fI R(1 S LOPE




~~-x





c( ~ di









c~ ~ a+ la



= 0 and y = Yo i





(

I

~I

1

1

I

I

I

r shy 1

bull bull bull r

I I

I

0

CASAG RAN DE - JIJ







l i I~JJ f Oil ~(H~

bull


FIG

~__~ lGLI__ _40_ __ ~ _ __

l

riG II b

-shy - FiG Ii J

II I

ill~ l























s I d

L

I

CAS GRAND E JIS









material
















i

I I

I

--shy

_

CASAGRA N DE 317


Y which I)



S S illl Cl --lt

I~ OJS~ 0

~ rO jgated for

pound ~ olutioll










0 leo rct iOIl

For proo f










nsforshy17

)y the He 12 used


dely timps

m In -hand limenshy

age- _

d proshya the

is an tial or

orlion a sa re smal



e b u l~

1)O$sess

made )IL1l In

tn3t a nrea-

CASAGRAr OE

~ Q( Q

I

~

t Q

~ ~ ~ ~ ~

I

~

I lt)

co 1

Q

-

- --




















I




---





Thee lo w TI lt



ISi rt It


da m and

seepage

CA onA middotDE Jll





l r---------~~--~

I A r

l~----------__--~~--~~~-=-~kl=-~~=~~~~~~~~~ o




shy







AlLOGY IhTHOD t I









313

r (


OJ


~on

t ad ~ shy


ty pe ill the

each on a lild J

I) t hI

r i ~oshy

1 due flow

l beshysuch ropi c

ldly such

CASAG RAN DE









A p PENDIX I



4h 4h ~a = k(I - = ~-

- a b

or kl C - =- (19 ) kl b

(I ca b = - - and - - = -shy


c ta n 3 kl - = -- = - (20)b lalla k


dtl tests r~(s out




ied in his


between n hip can

(1 9)

(20)


CASAGRAND JZS









- - -=1 i f

Cmiddot shy

r I

i middot







t r bull





CO S a sin (d - Wi)





n es ive rge

caw ion one

one

)oila ases


lmeshy

gtls in c by

1 one

(21)

327


(a)

c

(b)

o~~----~-r~shy

jJ = 90 + U)-c(




-





a) h)




5t re

equ tha i dln

gTq

(












CASAG RSDpound

329



6

8



r- t~middot r

1







ApPE nD II




21)0



~~~

+ p

r

(

19ths 9-10








y~-y I= - - shy

2yo 1i th Ee two

C-ISAGRND E JJI





I r- D shy





4



4J shy

I ~




1-1___ d ---_ - -shy-






1 -h J 1 bull bull




q = kh = 0539 kh 2Ii









middot







koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the







Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I

igtetwe-eo

In 193 1 o-dimenshy


~14)

lid the

J of one umputed

(15)

If rhe

~ +hl )~cisa d


( 16)



Surfaces






-~

- z

~ ~


LJIC fI R(1 S LOPE




~~-x





c( ~ di









c~ ~ a+ la



= 0 and y = Yo i





(

I

~I

1

1

I

I

I

r shy 1

bull bull bull r

I I

I

0

CASAG RAN DE - JIJ







l i I~JJ f Oil ~(H~

bull


FIG

~__~ lGLI__ _40_ __ ~ _ __

l

riG II b

-shy - FiG Ii J

II I

ill~ l























s I d

L

I

CAS GRAND E JIS









material
















i

I I

I

--shy

_

CASAGRA N DE 317


Y which I)



S S illl Cl --lt

I~ OJS~ 0

~ rO jgated for

pound ~ olutioll










0 leo rct iOIl

For proo f










nsforshy17

)y the He 12 used


dely timps

m In -hand limenshy

age- _

d proshya the

is an tial or

orlion a sa re smal



e b u l~

1)O$sess

made )IL1l In

tn3t a nrea-

CASAGRAr OE

~ Q( Q

I

~

t Q

~ ~ ~ ~ ~

I

~

I lt)

co 1

Q

-

- --




















I




---





Thee lo w TI lt



ISi rt It


da m and

seepage

CA onA middotDE Jll





l r---------~~--~

I A r

l~----------__--~~--~~~-=-~kl=-~~=~~~~~~~~~ o




shy







AlLOGY IhTHOD t I









313

r (


OJ


~on

t ad ~ shy


ty pe ill the

each on a lild J

I) t hI

r i ~oshy

1 due flow

l beshysuch ropi c

ldly such

CASAG RAN DE









A p PENDIX I



4h 4h ~a = k(I - = ~-

- a b

or kl C - =- (19 ) kl b

(I ca b = - - and - - = -shy


c ta n 3 kl - = -- = - (20)b lalla k


dtl tests r~(s out




ied in his


between n hip can

(1 9)

(20)


CASAGRAND JZS









- - -=1 i f

Cmiddot shy

r I

i middot







t r bull





CO S a sin (d - Wi)





n es ive rge

caw ion one

one

)oila ases


lmeshy

gtls in c by

1 one

(21)

327


(a)

c

(b)

o~~----~-r~shy

jJ = 90 + U)-c(




-





a) h)




5t re

equ tha i dln

gTq

(












CASAG RSDpound

329



6

8



r- t~middot r

1







ApPE nD II




21)0



~~~

+ p

r

(

19ths 9-10








y~-y I= - - shy

2yo 1i th Ee two

C-ISAGRND E JJI





I r- D shy





4



4J shy

I ~




1-1___ d ---_ - -shy-






1 -h J 1 bull bull




q = kh = 0539 kh 2Ii









middot







koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the







Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I




c( ~ di









c~ ~ a+ la



= 0 and y = Yo i





(

I

~I

1

1

I

I

I

r shy 1

bull bull bull r

I I

I

0

CASAG RAN DE - JIJ







l i I~JJ f Oil ~(H~

bull


FIG

~__~ lGLI__ _40_ __ ~ _ __

l

riG II b

-shy - FiG Ii J

II I

ill~ l























s I d

L

I

CAS GRAND E JIS









material
















i

I I

I

--shy

_

CASAGRA N DE 317


Y which I)



S S illl Cl --lt

I~ OJS~ 0

~ rO jgated for

pound ~ olutioll










0 leo rct iOIl

For proo f










nsforshy17

)y the He 12 used


dely timps

m In -hand limenshy

age- _

d proshya the

is an tial or

orlion a sa re smal



e b u l~

1)O$sess

made )IL1l In

tn3t a nrea-

CASAGRAr OE

~ Q( Q

I

~

t Q

~ ~ ~ ~ ~

I

~

I lt)

co 1

Q

-

- --




















I




---





Thee lo w TI lt



ISi rt It


da m and

seepage

CA onA middotDE Jll





l r---------~~--~

I A r

l~----------__--~~--~~~-=-~kl=-~~=~~~~~~~~~ o




shy







AlLOGY IhTHOD t I









313

r (


OJ


~on

t ad ~ shy


ty pe ill the

each on a lild J

I) t hI

r i ~oshy

1 due flow

l beshysuch ropi c

ldly such

CASAG RAN DE









A p PENDIX I



4h 4h ~a = k(I - = ~-

- a b

or kl C - =- (19 ) kl b

(I ca b = - - and - - = -shy


c ta n 3 kl - = -- = - (20)b lalla k


dtl tests r~(s out




ied in his


between n hip can

(1 9)

(20)


CASAGRAND JZS









- - -=1 i f

Cmiddot shy

r I

i middot







t r bull





CO S a sin (d - Wi)





n es ive rge

caw ion one

one

)oila ases


lmeshy

gtls in c by

1 one

(21)

327


(a)

c

(b)

o~~----~-r~shy

jJ = 90 + U)-c(




-





a) h)




5t re

equ tha i dln

gTq

(












CASAG RSDpound

329



6

8



r- t~middot r

1







ApPE nD II




21)0



~~~

+ p

r

(

19ths 9-10








y~-y I= - - shy

2yo 1i th Ee two

C-ISAGRND E JJI





I r- D shy





4



4J shy

I ~




1-1___ d ---_ - -shy-






1 -h J 1 bull bull




q = kh = 0539 kh 2Ii









middot







koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the







Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I

(

I

~I

1

1

I

I

I

r shy 1

bull bull bull r

I I

I

0

CASAG RAN DE - JIJ







l i I~JJ f Oil ~(H~

bull


FIG

~__~ lGLI__ _40_ __ ~ _ __

l

riG II b

-shy - FiG Ii J

II I

ill~ l























s I d

L

I

CAS GRAND E JIS









material
















i

I I

I

--shy

_

CASAGRA N DE 317


Y which I)



S S illl Cl --lt

I~ OJS~ 0

~ rO jgated for

pound ~ olutioll










0 leo rct iOIl

For proo f










nsforshy17

)y the He 12 used


dely timps

m In -hand limenshy

age- _

d proshya the

is an tial or

orlion a sa re smal



e b u l~

1)O$sess

made )IL1l In

tn3t a nrea-

CASAGRAr OE

~ Q( Q

I

~

t Q

~ ~ ~ ~ ~

I

~

I lt)

co 1

Q

-

- --




















I




---





Thee lo w TI lt



ISi rt It


da m and

seepage

CA onA middotDE Jll





l r---------~~--~

I A r

l~----------__--~~--~~~-=-~kl=-~~=~~~~~~~~~ o




shy







AlLOGY IhTHOD t I









313

r (


OJ


~on

t ad ~ shy


ty pe ill the

each on a lild J

I) t hI

r i ~oshy

1 due flow

l beshysuch ropi c

ldly such

CASAG RAN DE









A p PENDIX I



4h 4h ~a = k(I - = ~-

- a b

or kl C - =- (19 ) kl b

(I ca b = - - and - - = -shy


c ta n 3 kl - = -- = - (20)b lalla k


dtl tests r~(s out




ied in his


between n hip can

(1 9)

(20)


CASAGRAND JZS









- - -=1 i f

Cmiddot shy

r I

i middot







t r bull





CO S a sin (d - Wi)





n es ive rge

caw ion one

one

)oila ases


lmeshy

gtls in c by

1 one

(21)

327


(a)

c

(b)

o~~----~-r~shy

jJ = 90 + U)-c(




-





a) h)




5t re

equ tha i dln

gTq

(












CASAG RSDpound

329



6

8



r- t~middot r

1







ApPE nD II




21)0



~~~

+ p

r

(

19ths 9-10








y~-y I= - - shy

2yo 1i th Ee two

C-ISAGRND E JJI





I r- D shy





4



4J shy

I ~




1-1___ d ---_ - -shy-






1 -h J 1 bull bull




q = kh = 0539 kh 2Ii









middot







koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the







Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I




















s I d

L

I

CAS GRAND E JIS









material
















i

I I

I

--shy

_

CASAGRA N DE 317


Y which I)



S S illl Cl --lt

I~ OJS~ 0

~ rO jgated for

pound ~ olutioll










0 leo rct iOIl

For proo f










nsforshy17

)y the He 12 used


dely timps

m In -hand limenshy

age- _

d proshya the

is an tial or

orlion a sa re smal



e b u l~

1)O$sess

made )IL1l In

tn3t a nrea-

CASAGRAr OE

~ Q( Q

I

~

t Q

~ ~ ~ ~ ~

I

~

I lt)

co 1

Q

-

- --




















I




---





Thee lo w TI lt



ISi rt It


da m and

seepage

CA onA middotDE Jll





l r---------~~--~

I A r

l~----------__--~~--~~~-=-~kl=-~~=~~~~~~~~~ o




shy







AlLOGY IhTHOD t I









313

r (


OJ


~on

t ad ~ shy


ty pe ill the

each on a lild J

I) t hI

r i ~oshy

1 due flow

l beshysuch ropi c

ldly such

CASAG RAN DE









A p PENDIX I



4h 4h ~a = k(I - = ~-

- a b

or kl C - =- (19 ) kl b

(I ca b = - - and - - = -shy


c ta n 3 kl - = -- = - (20)b lalla k


dtl tests r~(s out




ied in his


between n hip can

(1 9)

(20)


CASAGRAND JZS









- - -=1 i f

Cmiddot shy

r I

i middot







t r bull





CO S a sin (d - Wi)





n es ive rge

caw ion one

one

)oila ases


lmeshy

gtls in c by

1 one

(21)

327


(a)

c

(b)

o~~----~-r~shy

jJ = 90 + U)-c(




-





a) h)




5t re

equ tha i dln

gTq

(












CASAG RSDpound

329



6

8



r- t~middot r

1







ApPE nD II




21)0



~~~

+ p

r

(

19ths 9-10








y~-y I= - - shy

2yo 1i th Ee two

C-ISAGRND E JJI





I r- D shy





4



4J shy

I ~




1-1___ d ---_ - -shy-






1 -h J 1 bull bull




q = kh = 0539 kh 2Ii









middot







koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the







Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I

I

CAS GRAND E JIS









material
















i

I I

I

--shy

_

CASAGRA N DE 317


Y which I)



S S illl Cl --lt

I~ OJS~ 0

~ rO jgated for

pound ~ olutioll










0 leo rct iOIl

For proo f










nsforshy17

)y the He 12 used


dely timps

m In -hand limenshy

age- _

d proshya the

is an tial or

orlion a sa re smal



e b u l~

1)O$sess

made )IL1l In

tn3t a nrea-

CASAGRAr OE

~ Q( Q

I

~

t Q

~ ~ ~ ~ ~

I

~

I lt)

co 1

Q

-

- --




















I




---





Thee lo w TI lt



ISi rt It


da m and

seepage

CA onA middotDE Jll





l r---------~~--~

I A r

l~----------__--~~--~~~-=-~kl=-~~=~~~~~~~~~ o




shy







AlLOGY IhTHOD t I









313

r (


OJ


~on

t ad ~ shy


ty pe ill the

each on a lild J

I) t hI

r i ~oshy

1 due flow

l beshysuch ropi c

ldly such

CASAG RAN DE









A p PENDIX I



4h 4h ~a = k(I - = ~-

- a b

or kl C - =- (19 ) kl b

(I ca b = - - and - - = -shy


c ta n 3 kl - = -- = - (20)b lalla k


dtl tests r~(s out




ied in his


between n hip can

(1 9)

(20)


CASAGRAND JZS









- - -=1 i f

Cmiddot shy

r I

i middot







t r bull





CO S a sin (d - Wi)





n es ive rge

caw ion one

one

)oila ases


lmeshy

gtls in c by

1 one

(21)

327


(a)

c

(b)

o~~----~-r~shy

jJ = 90 + U)-c(




-





a) h)




5t re

equ tha i dln

gTq

(












CASAG RSDpound

329



6

8



r- t~middot r

1







ApPE nD II




21)0



~~~

+ p

r

(

19ths 9-10








y~-y I= - - shy

2yo 1i th Ee two

C-ISAGRND E JJI





I r- D shy





4



4J shy

I ~




1-1___ d ---_ - -shy-






1 -h J 1 bull bull




q = kh = 0539 kh 2Ii









middot







koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the







Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I










i

I I

I

--shy

_

CASAGRA N DE 317


Y which I)



S S illl Cl --lt

I~ OJS~ 0

~ rO jgated for

pound ~ olutioll










0 leo rct iOIl

For proo f










nsforshy17

)y the He 12 used


dely timps

m In -hand limenshy

age- _

d proshya the

is an tial or

orlion a sa re smal



e b u l~

1)O$sess

made )IL1l In

tn3t a nrea-

CASAGRAr OE

~ Q( Q

I

~

t Q

~ ~ ~ ~ ~

I

~

I lt)

co 1

Q

-

- --




















I




---





Thee lo w TI lt



ISi rt It


da m and

seepage

CA onA middotDE Jll





l r---------~~--~

I A r

l~----------__--~~--~~~-=-~kl=-~~=~~~~~~~~~ o




shy







AlLOGY IhTHOD t I









313

r (


OJ


~on

t ad ~ shy


ty pe ill the

each on a lild J

I) t hI

r i ~oshy

1 due flow

l beshysuch ropi c

ldly such

CASAG RAN DE









A p PENDIX I



4h 4h ~a = k(I - = ~-

- a b

or kl C - =- (19 ) kl b

(I ca b = - - and - - = -shy


c ta n 3 kl - = -- = - (20)b lalla k


dtl tests r~(s out




ied in his


between n hip can

(1 9)

(20)


CASAGRAND JZS









- - -=1 i f

Cmiddot shy

r I

i middot







t r bull





CO S a sin (d - Wi)





n es ive rge

caw ion one

one

)oila ases


lmeshy

gtls in c by

1 one

(21)

327


(a)

c

(b)

o~~----~-r~shy

jJ = 90 + U)-c(




-





a) h)




5t re

equ tha i dln

gTq

(












CASAG RSDpound

329



6

8



r- t~middot r

1







ApPE nD II




21)0



~~~

+ p

r

(

19ths 9-10








y~-y I= - - shy

2yo 1i th Ee two

C-ISAGRND E JJI





I r- D shy





4



4J shy

I ~




1-1___ d ---_ - -shy-






1 -h J 1 bull bull




q = kh = 0539 kh 2Ii









middot







koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the







Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I

_

CASAGRA N DE 317


Y which I)



S S illl Cl --lt

I~ OJS~ 0

~ rO jgated for

pound ~ olutioll










0 leo rct iOIl

For proo f










nsforshy17

)y the He 12 used


dely timps

m In -hand limenshy

age- _

d proshya the

is an tial or

orlion a sa re smal



e b u l~

1)O$sess

made )IL1l In

tn3t a nrea-

CASAGRAr OE

~ Q( Q

I

~

t Q

~ ~ ~ ~ ~

I

~

I lt)

co 1

Q

-

- --




















I




---





Thee lo w TI lt



ISi rt It


da m and

seepage

CA onA middotDE Jll





l r---------~~--~

I A r

l~----------__--~~--~~~-=-~kl=-~~=~~~~~~~~~ o




shy







AlLOGY IhTHOD t I









313

r (


OJ


~on

t ad ~ shy


ty pe ill the

each on a lild J

I) t hI

r i ~oshy

1 due flow

l beshysuch ropi c

ldly such

CASAG RAN DE









A p PENDIX I



4h 4h ~a = k(I - = ~-

- a b

or kl C - =- (19 ) kl b

(I ca b = - - and - - = -shy


c ta n 3 kl - = -- = - (20)b lalla k


dtl tests r~(s out




ied in his


between n hip can

(1 9)

(20)


CASAGRAND JZS









- - -=1 i f

Cmiddot shy

r I

i middot







t r bull





CO S a sin (d - Wi)





n es ive rge

caw ion one

one

)oila ases


lmeshy

gtls in c by

1 one

(21)

327


(a)

c

(b)

o~~----~-r~shy

jJ = 90 + U)-c(




-





a) h)




5t re

equ tha i dln

gTq

(












CASAG RSDpound

329



6

8



r- t~middot r

1







ApPE nD II




21)0



~~~

+ p

r

(

19ths 9-10








y~-y I= - - shy

2yo 1i th Ee two

C-ISAGRND E JJI





I r- D shy





4



4J shy

I ~




1-1___ d ---_ - -shy-






1 -h J 1 bull bull




q = kh = 0539 kh 2Ii









middot







koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the







Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I










nsforshy17

)y the He 12 used


dely timps

m In -hand limenshy

age- _

d proshya the

is an tial or

orlion a sa re smal



e b u l~

1)O$sess

made )IL1l In

tn3t a nrea-

CASAGRAr OE

~ Q( Q

I

~

t Q

~ ~ ~ ~ ~

I

~

I lt)

co 1

Q

-

- --




















I




---





Thee lo w TI lt



ISi rt It


da m and

seepage

CA onA middotDE Jll





l r---------~~--~

I A r

l~----------__--~~--~~~-=-~kl=-~~=~~~~~~~~~ o




shy







AlLOGY IhTHOD t I









313

r (


OJ


~on

t ad ~ shy


ty pe ill the

each on a lild J

I) t hI

r i ~oshy

1 due flow

l beshysuch ropi c

ldly such

CASAG RAN DE









A p PENDIX I



4h 4h ~a = k(I - = ~-

- a b

or kl C - =- (19 ) kl b

(I ca b = - - and - - = -shy


c ta n 3 kl - = -- = - (20)b lalla k


dtl tests r~(s out




ied in his


between n hip can

(1 9)

(20)


CASAGRAND JZS









- - -=1 i f

Cmiddot shy

r I

i middot







t r bull





CO S a sin (d - Wi)





n es ive rge

caw ion one

one

)oila ases


lmeshy

gtls in c by

1 one

(21)

327


(a)

c

(b)

o~~----~-r~shy

jJ = 90 + U)-c(




-





a) h)




5t re

equ tha i dln

gTq

(












CASAG RSDpound

329



6

8



r- t~middot r

1







ApPE nD II




21)0



~~~

+ p

r

(

19ths 9-10








y~-y I= - - shy

2yo 1i th Ee two

C-ISAGRND E JJI





I r- D shy





4



4J shy

I ~




1-1___ d ---_ - -shy-






1 -h J 1 bull bull




q = kh = 0539 kh 2Ii









middot







koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the







Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I

nsforshy17

)y the He 12 used


dely timps

m In -hand limenshy

age- _

d proshya the

is an tial or

orlion a sa re smal



e b u l~

1)O$sess

made )IL1l In

tn3t a nrea-

CASAGRAr OE

~ Q( Q

I

~

t Q

~ ~ ~ ~ ~

I

~

I lt)

co 1

Q

-

- --




















I




---





Thee lo w TI lt



ISi rt It


da m and

seepage

CA onA middotDE Jll





l r---------~~--~

I A r

l~----------__--~~--~~~-=-~kl=-~~=~~~~~~~~~ o




shy







AlLOGY IhTHOD t I









313

r (


OJ


~on

t ad ~ shy


ty pe ill the

each on a lild J

I) t hI

r i ~oshy

1 due flow

l beshysuch ropi c

ldly such

CASAG RAN DE









A p PENDIX I



4h 4h ~a = k(I - = ~-

- a b

or kl C - =- (19 ) kl b

(I ca b = - - and - - = -shy


c ta n 3 kl - = -- = - (20)b lalla k


dtl tests r~(s out




ied in his


between n hip can

(1 9)

(20)


CASAGRAND JZS









- - -=1 i f

Cmiddot shy

r I

i middot







t r bull





CO S a sin (d - Wi)





n es ive rge

caw ion one

one

)oila ases


lmeshy

gtls in c by

1 one

(21)

327


(a)

c

(b)

o~~----~-r~shy

jJ = 90 + U)-c(




-





a) h)




5t re

equ tha i dln

gTq

(












CASAG RSDpound

329



6

8



r- t~middot r

1







ApPE nD II




21)0



~~~

+ p

r

(

19ths 9-10








y~-y I= - - shy

2yo 1i th Ee two

C-ISAGRND E JJI





I r- D shy





4



4J shy

I ~




1-1___ d ---_ - -shy-






1 -h J 1 bull bull




q = kh = 0539 kh 2Ii









middot







koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the







Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I

- --




















I




---





Thee lo w TI lt



ISi rt It


da m and

seepage

CA onA middotDE Jll





l r---------~~--~

I A r

l~----------__--~~--~~~-=-~kl=-~~=~~~~~~~~~ o




shy







AlLOGY IhTHOD t I









313

r (


OJ


~on

t ad ~ shy


ty pe ill the

each on a lild J

I) t hI

r i ~oshy

1 due flow

l beshysuch ropi c

ldly such

CASAG RAN DE









A p PENDIX I



4h 4h ~a = k(I - = ~-

- a b

or kl C - =- (19 ) kl b

(I ca b = - - and - - = -shy


c ta n 3 kl - = -- = - (20)b lalla k


dtl tests r~(s out




ied in his


between n hip can

(1 9)

(20)


CASAGRAND JZS









- - -=1 i f

Cmiddot shy

r I

i middot







t r bull





CO S a sin (d - Wi)





n es ive rge

caw ion one

one

)oila ases


lmeshy

gtls in c by

1 one

(21)

327


(a)

c

(b)

o~~----~-r~shy

jJ = 90 + U)-c(




-





a) h)




5t re

equ tha i dln

gTq

(












CASAG RSDpound

329



6

8



r- t~middot r

1







ApPE nD II




21)0



~~~

+ p

r

(

19ths 9-10








y~-y I= - - shy

2yo 1i th Ee two

C-ISAGRND E JJI





I r- D shy





4



4J shy

I ~




1-1___ d ---_ - -shy-






1 -h J 1 bull bull




q = kh = 0539 kh 2Ii









middot







koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the







Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I

---





Thee lo w TI lt



ISi rt It


da m and

seepage

CA onA middotDE Jll





l r---------~~--~

I A r

l~----------__--~~--~~~-=-~kl=-~~=~~~~~~~~~ o




shy







AlLOGY IhTHOD t I









313

r (


OJ


~on

t ad ~ shy


ty pe ill the

each on a lild J

I) t hI

r i ~oshy

1 due flow

l beshysuch ropi c

ldly such

CASAG RAN DE









A p PENDIX I



4h 4h ~a = k(I - = ~-

- a b

or kl C - =- (19 ) kl b

(I ca b = - - and - - = -shy


c ta n 3 kl - = -- = - (20)b lalla k


dtl tests r~(s out




ied in his


between n hip can

(1 9)

(20)


CASAGRAND JZS









- - -=1 i f

Cmiddot shy

r I

i middot







t r bull





CO S a sin (d - Wi)





n es ive rge

caw ion one

one

)oila ases


lmeshy

gtls in c by

1 one

(21)

327


(a)

c

(b)

o~~----~-r~shy

jJ = 90 + U)-c(




-





a) h)




5t re

equ tha i dln

gTq

(












CASAG RSDpound

329



6

8



r- t~middot r

1







ApPE nD II




21)0



~~~

+ p

r

(

19ths 9-10








y~-y I= - - shy

2yo 1i th Ee two

C-ISAGRND E JJI





I r- D shy





4



4J shy

I ~




1-1___ d ---_ - -shy-






1 -h J 1 bull bull




q = kh = 0539 kh 2Ii









middot







koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the







Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I







AlLOGY IhTHOD t I









313

r (


OJ


~on

t ad ~ shy


ty pe ill the

each on a lild J

I) t hI

r i ~oshy

1 due flow

l beshysuch ropi c

ldly such

CASAG RAN DE









A p PENDIX I



4h 4h ~a = k(I - = ~-

- a b

or kl C - =- (19 ) kl b

(I ca b = - - and - - = -shy


c ta n 3 kl - = -- = - (20)b lalla k


dtl tests r~(s out




ied in his


between n hip can

(1 9)

(20)


CASAGRAND JZS









- - -=1 i f

Cmiddot shy

r I

i middot







t r bull





CO S a sin (d - Wi)





n es ive rge

caw ion one

one

)oila ases


lmeshy

gtls in c by

1 one

(21)

327


(a)

c

(b)

o~~----~-r~shy

jJ = 90 + U)-c(




-





a) h)




5t re

equ tha i dln

gTq

(












CASAG RSDpound

329



6

8



r- t~middot r

1







ApPE nD II




21)0



~~~

+ p

r

(

19ths 9-10








y~-y I= - - shy

2yo 1i th Ee two

C-ISAGRND E JJI





I r- D shy





4



4J shy

I ~




1-1___ d ---_ - -shy-






1 -h J 1 bull bull




q = kh = 0539 kh 2Ii









middot







koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the







Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I

313

r (


OJ


~on

t ad ~ shy


ty pe ill the

each on a lild J

I) t hI

r i ~oshy

1 due flow

l beshysuch ropi c

ldly such

CASAG RAN DE









A p PENDIX I



4h 4h ~a = k(I - = ~-

- a b

or kl C - =- (19 ) kl b

(I ca b = - - and - - = -shy


c ta n 3 kl - = -- = - (20)b lalla k


dtl tests r~(s out




ied in his


between n hip can

(1 9)

(20)


CASAGRAND JZS









- - -=1 i f

Cmiddot shy

r I

i middot







t r bull





CO S a sin (d - Wi)





n es ive rge

caw ion one

one

)oila ases


lmeshy

gtls in c by

1 one

(21)

327


(a)

c

(b)

o~~----~-r~shy

jJ = 90 + U)-c(




-





a) h)




5t re

equ tha i dln

gTq

(












CASAG RSDpound

329



6

8



r- t~middot r

1







ApPE nD II




21)0



~~~

+ p

r

(

19ths 9-10








y~-y I= - - shy

2yo 1i th Ee two

C-ISAGRND E JJI





I r- D shy





4



4J shy

I ~




1-1___ d ---_ - -shy-






1 -h J 1 bull bull




q = kh = 0539 kh 2Ii









middot







koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the







Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I




A p PENDIX I



4h 4h ~a = k(I - = ~-

- a b

or kl C - =- (19 ) kl b

(I ca b = - - and - - = -shy


c ta n 3 kl - = -- = - (20)b lalla k


dtl tests r~(s out




ied in his


between n hip can

(1 9)

(20)


CASAGRAND JZS









- - -=1 i f

Cmiddot shy

r I

i middot







t r bull





CO S a sin (d - Wi)





n es ive rge

caw ion one

one

)oila ases


lmeshy

gtls in c by

1 one

(21)

327


(a)

c

(b)

o~~----~-r~shy

jJ = 90 + U)-c(




-





a) h)




5t re

equ tha i dln

gTq

(












CASAG RSDpound

329



6

8



r- t~middot r

1







ApPE nD II




21)0



~~~

+ p

r

(

19ths 9-10








y~-y I= - - shy

2yo 1i th Ee two

C-ISAGRND E JJI





I r- D shy





4



4J shy

I ~




1-1___ d ---_ - -shy-






1 -h J 1 bull bull




q = kh = 0539 kh 2Ii









middot







koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the







Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I

dtl tests r~(s out




ied in his


between n hip can

(1 9)

(20)


CASAGRAND JZS









- - -=1 i f

Cmiddot shy

r I

i middot







t r bull





CO S a sin (d - Wi)





n es ive rge

caw ion one

one

)oila ases


lmeshy

gtls in c by

1 one

(21)

327


(a)

c

(b)

o~~----~-r~shy

jJ = 90 + U)-c(




-





a) h)




5t re

equ tha i dln

gTq

(












CASAG RSDpound

329



6

8



r- t~middot r

1







ApPE nD II




21)0



~~~

+ p

r

(

19ths 9-10








y~-y I= - - shy

2yo 1i th Ee two

C-ISAGRND E JJI





I r- D shy





4



4J shy

I ~




1-1___ d ---_ - -shy-






1 -h J 1 bull bull




q = kh = 0539 kh 2Ii









middot







koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the







Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I







t r bull





CO S a sin (d - Wi)





n es ive rge

caw ion one

one

)oila ases


lmeshy

gtls in c by

1 one

(21)

327


(a)

c

(b)

o~~----~-r~shy

jJ = 90 + U)-c(




-





a) h)




5t re

equ tha i dln

gTq

(












CASAG RSDpound

329



6

8



r- t~middot r

1







ApPE nD II




21)0



~~~

+ p

r

(

19ths 9-10








y~-y I= - - shy

2yo 1i th Ee two

C-ISAGRND E JJI





I r- D shy





4



4J shy

I ~




1-1___ d ---_ - -shy-






1 -h J 1 bull bull




q = kh = 0539 kh 2Ii









middot







koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the







Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I

n es ive rge

caw ion one

one

)oila ases


lmeshy

gtls in c by

1 one

(21)

327


(a)

c

(b)

o~~----~-r~shy

jJ = 90 + U)-c(




-





a) h)




5t re

equ tha i dln

gTq

(












CASAG RSDpound

329



6

8



r- t~middot r

1







ApPE nD II




21)0



~~~

+ p

r

(

19ths 9-10








y~-y I= - - shy

2yo 1i th Ee two

C-ISAGRND E JJI





I r- D shy





4



4J shy

I ~




1-1___ d ---_ - -shy-






1 -h J 1 bull bull




q = kh = 0539 kh 2Ii









middot







koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the







Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I





a) h)




5t re

equ tha i dln

gTq

(












CASAG RSDpound

329



6

8



r- t~middot r

1







ApPE nD II




21)0



~~~

+ p

r

(

19ths 9-10








y~-y I= - - shy

2yo 1i th Ee two

C-ISAGRND E JJI





I r- D shy





4



4J shy

I ~




1-1___ d ---_ - -shy-






1 -h J 1 bull bull




q = kh = 0539 kh 2Ii









middot







koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the







Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I












CASAG RSDpound

329



6

8



r- t~middot r

1







ApPE nD II




21)0



~~~

+ p

r

(

19ths 9-10








y~-y I= - - shy

2yo 1i th Ee two

C-ISAGRND E JJI





I r- D shy





4



4J shy

I ~




1-1___ d ---_ - -shy-






1 -h J 1 bull bull




q = kh = 0539 kh 2Ii









middot







koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the







Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I

1







ApPE nD II




21)0



~~~

+ p

r

(

19ths 9-10








y~-y I= - - shy

2yo 1i th Ee two

C-ISAGRND E JJI





I r- D shy





4



4J shy

I ~




1-1___ d ---_ - -shy-






1 -h J 1 bull bull




q = kh = 0539 kh 2Ii









middot







koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the







Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I

r

(

19ths 9-10








y~-y I= - - shy

2yo 1i th Ee two

C-ISAGRND E JJI





I r- D shy





4



4J shy

I ~




1-1___ d ---_ - -shy-






1 -h J 1 bull bull




q = kh = 0539 kh 2Ii









middot







koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the







Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I

4J shy

I ~




1-1___ d ---_ - -shy-






1 -h J 1 bull bull




q = kh = 0539 kh 2Ii









middot







koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the







Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I









middot







koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the







Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I

middot







koeq

8id( ~

AUgmiddot

l og

IS61

rler Jul

the







Umiddot

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I

CA SA GRA-iOE _ 335

BI BLIOGRAPHY

























I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I













B

Vol




pro) of ( si t (

and

to t

seq c T wt-

of sh ~

oil i nl Ol

s t r

I

Documents

Casagrande - FILTRACION EN PRESAS