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\Vater H)·tlrologp .
I':nginl,,>rlng '273. Ad \' anced Topics i n
IUVTOjJOt; 6vQo..onit ",""Ii
Plp~loypa.[Qt;
T6 np66A.Tl)..LQ tOC Boussin esq, f1tO( tfte; te,larocr£co; (1) ICQl
'tellV Pon911t UcGlv O\Iv9ll(&v (2) p€ d = 0, r:XEI t1tlO"TlC; A.u9ft
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n'}tWC; tftt;; i:pyao(a<; TO(} Boussinesq. '0 L.D. Dumm 'to ftoc;
1954 lCal o{ J an van S chilfgaarde I("al A.<>ltcoi TO ho(; 1956 m:l.
pouainoav .i]v A.OO"LV toO npo p)..TtlJQtoc;- tfie; ~c,. ( I) Ka t Tlllv
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pic; vii dVCLCpE'p900v E{e; tilV tpyacriav tOO Bouss in esq . '0 Jan
lSI
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Polubarinova·Kochina TO 1952
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(II) dc; n)v
'H apxll('I) ouvefJ Kll H(s,o)= I , F(I)= 1 Un:OO£ II(VUEL On
C~= I. Kat rJx; h TQUTOU fJ t;;. (2?) Y1VEtUl:
I
c.l
t;;,
r('I
I(al B (a,b) ftVfl1!:POOW1tt;UE l n)v n: A.i}Pll Be ta - cruvliPTllolV
J
(26) Kat
PQ.vo~v:
(20)
(complele Beta functioo) 6pl<:o~v T) urt6
B( a, b) -
e.;.
'AV'CIKaelOtGlvnc; 'tal.;
[1-I,(f, ~)]
(~J'
k=
153
Xpovu,d
(30)
' H t;;. (30) SUVo.1C1.t va. 7..PT)O"I)Jon:oITlStl 016. TOV O1tOA.o), 10"110V
Tiie; 6,n:o(J-rCz.O'ewt, TWV O'-rPCI.),),lO'W(GlV n: 1lA.OO'WAfjVWV.
'H -r1J.LfJ -roC A oiSel01 {)ItO 10C n(VUKOC; I
alt6 tnV €i;icrwO"lv:
i1
'toO LX . 2 ~ I(o.i
(31)
,I
14
niNA _
f
A
dlJio
A
dlno
A
0. ';;0
0, 90.8
0,60
0.,782
0. 80
0. 575
0, 972
0.47
0. 898
0, 67
0. 7d6
0. 82
0,548
0,2"
0. 967
0. .44
0,888
0. 64
0. 7S0
0,84
0. 5/9
0,998
0,26
0,962
0, ,,6
0. 817
0. 66
0. 737
0,86
0.487
0.08
0 .996
0.28
0. 966
o.p8
0. 865
0.68
OJI8
0.88
0.453
0,10
o. 994
o. 90
0. 948
o_50
0.855
070.
0.692
0. 90
0. ,&/1./
0.12
0. 991
0. 92
0. 9p7
aS2
0, 840
0.7?
467/
0. 92
0,372
0,14
0. 988
0,9/;
0. 933
0. 6.4
0,826
47p
0. 660
0,94
0,32/
0.16
0985
0.86
0,926
17. 56
0. 8/8
(J 76
0.626
CJ.
0. 18
(J,981
0. 38
O,9/T
aS8
0. 797
0. 78
0,602
0, 98
0, /84
!. 00
0.000
d~o
A
dl"o
A
0, 00
1,000
0,70
0,977
0, 0.2
0,999
0.22
O.OLi
0.. 999
0,06
dih#
96
0. 260
254
TltXytkG
lO r---~~----------------------~
0,9
0.8F-______________
'ATTplklOC; 1969
XpOYUdl
't6 ~aOoc; l,SIl . 0.1[6 T\'jC; trrupavda.c; lO\) to(l.(poo<; £vl6c; t 5
U£PGlV (neptoooc; J.1E1oJ;U buo 6.pOEUcrr.mV);
AoO"lC;: 'A1to to. OEOOJ,ltVo. TOO dx; t'.i:vw nppA..illlOtOt; ta. crOJ,I­
POA.o. TU)V u"lO'COaEWV fxouv tat; ICOtroOl 1LJ,lat;:
~
d = 7,5-2,5 =
K - 0,366
5,0~. ~/~~epa S - 0 ,08 Yo = 2,5- 1,0= 1,511· Y _ 2,5_1,5= I ,O~. ~6
""•
"
•
t-::.
1\­
p.,
h. = 7,5- 1,0=
Ix. 2
~.
(,;It" Tlj./Qt rtN' A
"- 03
eK
TOU
11'4/
..!L
a =
~.
6,5~, = 7,5-1,5=6.0~ .
h
0, 127~ .
J)
.Il / l'a}(,Of
0.7
1. np6'ltO(f 1(61(1.01; bnOA.o1'oJ,llbv
0.1
a) 'A1t6 10. o£5oJ,l£va
CntOA.oY{~OJ.1EV 'tnv 'tLJ,lnV toO ~o
Kat )(0..
t6ntv un6 to 1:X. 3 £upiaKOtJ,.6v tnV 'tLJ.1Tlv A.
hd -
5
- 5 = 0,?69.
6,
•
rou
d.
Ii.
'H t~. (31) SicEl KOfel rrpootYYlO"lV
'toO 3%.
p> ' A1t6 TTlV tl;lawcnv (30) A..aJ..l~aVOU£ ~ tnv npon'lv leata n:poa­
ErtlO'IV unoato.O"lv
t"('.Ul~ ~
A.nSae; Ilucponpov
.6u';pOWOUi o+«A.,aaTOf:; }t.6y"tJ ouy,,"la&:wfi VpalA.... Qv pof\.;
lle:SoJ,.LAvou c'hl 1) lhocpoPIICfI t~iawO'u; (\) Sev !crXUE~ rr)..'lO'lO\l
ICA.ElCJ1"G:lv 6:ywy&v C"'tpay-y(aewr; ).6y~ 11\<; CJ1YYICA.(OEO:X;
ffie; pol'lc; (cSlSla(T1aTO~ pol'll 6 Jan van SchiHgaaT'de OneSE~sE
1'1'r\l xpflcnv tol) (<laoSuva~o-.l pa801.)(;}) 'TOO S. E. Hooghuudl
It!>&; avmwfacrtaaLY 'toO n:paYIlQtllcOO P6:8ou<; ~h6 'tn'll CLOp·
9roaw TO'\} il1tEIO'£PXO).lEVQU crqul\J.1a'(~ . '0 W. T. Moody
'(6:)\1
"aTa
TO hoc; 1966 t~TiYQYE J.llav 6.1tA.OUatEJ)Q" Kal
"(lOW tt,IO'(OOlV Ola n)v 6,opO<OOIV m>YKA.(crE~:
1tpocrEy­
d
d [-8
+ -X
lor
°
~
d) In ( -IX
6nO'u d t{vm TO tuoSuvaJ.lOV pa90C;;,
voc; Kat
11'1V
r Elvat
tJ.{Q O'UVO:P'tllO"lt; 'tot)
Q
=
5
95.3 = 0,0525
~'Ol
d
X
< 0, 3
'Ano tnV t!;iaroow (33)
r _ 3,5:'
1, 6 (0, 0525) + 2 (0 ,0525)' = 3,472
d) 'A1tO t1\v e.!;lomOlv (n) '" 6no{0. \m'0 1l0PIPT'lv OEICa.011C00
A..010pi9J.10U 1patpEtal:
[1 +
-=
elvat ft (h:tic;; TOt! aO»),.~ ­
Xd 1\
d
c) y
(32)
X ':; 0,3
A
2(0617)[1 ,11 5(0,366) (5 +~ ) (5 + 1 ,5)15]'1. = 953
,
0,08 (6,5 - 6,0)
, ~
d
r-­ ] - ' ,
d
x=
d
A
-d- = [ 1
'An6 to 'EX , 2 £uplaKollEv A = 0,617
(0,0525) (5,86iOg ( 0,;27) - 3,472)
= 0,764
..Apa
01to{o. O{OSOo.t o.1tO
r'
(32.)
d . 3,82 Il
Elvat t6 laoouvoJ.10v pa80c;.
1. Ac6npoc; I(lnO"oc; b1toA..01lCrp&v
t~ ia(ool v:
r _ 3 ,55-1 ,6(-~-)+2(~)'
d
dlO: 't1J.16~ 'toO X
> 0,3,
(33)
d
'H ttllTl toO [0'08uvaJ.1ou jlo.0oUC; = 3,821l., 1\ EupE9£[cro EtC;
TOV 1tpOltOV ICU!CA.OV unOAOYlatullV, XPTlalIlOn:ol£[tm ffi<; TLJ.1n
't0C) d f.iC; T1'IV tl;la<oow (30) ci>; !Co.tooOt :
1\ Ko.ta 1tpoa£'YYIO"IIJ tl;{awmt; tOC)
h
Hooghoudt croVtCTtd1al, i\tOl
8
Xd - [ ~[In
d + y = 3,82+ 1,0= 4,82
h, = d + y. = 3,82 + 1,5 = 5,32
(X)
"
-1,15] ]-'
(34)
"E+op\-1oyfrt ~4j 9&:wp\o4j de.; npalCTucov npbPAlwa
'y.no8taCOJ.1EV on O'tpa1YIO"TLICOI 1t1lA.oO'OOA.!'Iv~ Olo.J.1tTPOU
10 lV'taG':\v (25,4 tIC.) 1t p6ICElTal va tonoOETTlBotiv de; pa90C;
2,5 ~tpUlV tino 1~t; £1tUpavE(a..r; roO tMq>out;. 'H toalpOA,oyllCl'\
~pEuva EOOOO'E ta ":(1TC1l81 O"rOlXdo : a) To a 6lanepa<1'tov aP1L­
A..IICOV aTp(DJJo. eOplO'K'ETOI eit; p6.90e; 7,SJ,I. o.n;O 'tile;; trtupavEiac;
tot} t&alpo~ . b) 'H i>opauA..uo\ 6,1001l1.l0rrp; to\) tOUIpOUC; Elvat
ICOTo. ~O'OV 6pov 0,366 Il£tpa n,v ftJ,ltpav (24copov). c) 'H
eLoLKTl aTpa.Y110'tuC~ hcav6tl1<; 'toC) £o6.tpOll~ elvat 0,08 Kat'
61KOV nOprov.
no lo. np£n£l vd elval 1'1 Q1t6mame; IlE'tal;o t&v 1t1lA..oO'OOA..l'lvwv
oOtoo<; OO'tE 1'1 ()1tcry£la ataBIlTl 'tOt) ubuHJo<;, 1\ EUpLaKotJ.f:VTl £ie;
f}a9oc; 1,011. Qno tflt; tnupov£lUC; TOO !oO:tpOUt;, va. )(attj"Ou elc;
J
0)
h.
.
b)' An6 t!'lv tl;lawaw (30) kaJ,lpttvoj.l.Ev t!'lv OEUttpa.V )((1Ta
1tpoO'£'Y1lollJ ttJ.1Tlv TI"] C; aTtOato.aEw;
X =
2
(0673) [ 1,11 5(0,366) (4,82) (5,32)15 ]'" ~ 84 4
, ~
,
0,08 (5,32 _ 4,82)
'H Qvoottp(l) l'lJ.1n TOO X XPTlmJ.10not£Tt(11 liu1 t"'V £Up£O'lV 1fic;
vto( 't1J.1!'1C; toO laoouvo.J..lOU paOouc;
c)
d
X
5
= 84,4 - 0,05925,
~TOI
d
X
< 0,3
'An6 T~V t~[""""V (33) r = 3,55 - 1,6 (0,05925) + 2 (0,05925)' = 3,462 155
'Arrp[).to.; 1969
d) 'A1t6 trlV &E,iou)(JLV (320)
{~[ t
")
+ 10,05925) (5,8610g ( 0, ;27 ) -
3,,62) ]-'
d=
3. Tplta<; K6K).O~ t.1tO).OYIC1JJWV
d
O1Jvex,(aw~ev 'l'OUo:; U1tOA.oY la~o i.le;, 1'\ ttJlT! '1'00 X Sci I-ISta­
jl)..llOtt 1-I0VOv l(atO ~EPtI(U tKatoO'tCt. 'no:; ~I( 1'OU'1'OU, ~ ~"'tou­
J,1£Vl1 tl~r1 't'i)<; 01tOCtaaEWO:; ~6taE.u T&V O''l'paYYIO'tll(&v nll)..o­
O"WA.ti vwv 90. Elval 83 J1ttpa .
'Ea\!
. H nil T't TOC IOOouvafl,Ou tkt90uc; = 3.7 J il £i.lpegetO'Cl elo:; tOV
oeuT£pov KUI(A.OV lmo;"oYlO'Il&V XPllO'lllOrtOl!dtClI roo:; Tlj.l1'l TOO
d ell; niv tE,icl'(oolv (30) ~ l(a:tw91 :
h - d+ y ~ 3,71 + 1,0 = 4,71
h. - d + y = 3,71 + 1,5 ~ 5,21
·0).01 o{ ovu)tepw
p19j.l l1C00 1(0VO\lOe;.
d
371
.
• ) -h = -5'2 =0 ,712. 'Ano TOl:X. 2 eup[O'Ko~V A=O,681
, I
'H OVOOtepW ttllT! rOi) X x'PllO'LIlOJrOIEitClI Ota Tr,v Ei.lp£O'w r1')o:;
vtao:; tlJ.l~O:; toO iaoOUVClIlOl) Pc't8ouo:;
5
= 80,6
'Ano tr,v
I
~
d
< 0.3
~ 0,0598 ,~TO' X
~[O"waLv
2. 'EK ti)e; tlVaAUtllCf'lO:; )..uaf..We; eupt9ll j.lia lE,{O'u)(lI<; (rimoo:;),
Sui tov uno)..oYlO'j.lOV tf'lc; Q.rtoO''tQaec:,o.; I-Ietasu
O"TpaYYI­
0'1'l1((bv l()..uO'tCUv dywyGlv, UrtO '1'OlauTll v 1-I0PQ>1\v &O't E v6 bU­
vatQI va x'PllO'lllonolll9'i'\ eOICOA.WI; S~a '1'r,v tl(1tOV'IlO'I V 1l£4·
T&V O''1'pa.rf IO'tLKGw OIIC'6wv.
,6)"
(33)
~
3.55 - 1,6(0,0598) + 2(0,0598)'
3,4615
3, 'EttttoJi it tsaywyr, -rile; tslaroatWo:; TOO Boussinesq (BA..
naPQP.T1Ila) Paai(.e.at Ertl tG>v rtapaooxGlv '1'&V Dupuit lCal
F'orchh eimer, to U7telO'I:':PXOI-I£VOv aq:ul)..lla, A.6Y4l tf'le; O'uYl()..i­
O'Ew.; Tillv ypalll-l&v pof'l" olOp90ihai eto:; '1'TlV rrapoOaav civll­
A.l)O IV tf.'t f}OI)9EiQ. tGlv tsLO'ooaEwv '1'00 «(O'oOUVQj.lOU pa90UQ)
toO H ooghoudl roo:; aQ'1'ai ex,ouv 6::rrA.onot,,9fl uno 1'00 Moody
0) 'An6 l'1'\v tslawO"tV (32a)
%=
[1 + 10,0598) ( 5,86 log ( 0,;27 ) - 3,461 5)
r
=
~0 . n9
to tto.; 1966.
•Apa d = (0,139) 5 = 3,695 Il. Elvcn '1'0 vtov {O"OOUVClI-IOV
4.
TtfQPl'o~
EXOUV tKteM:a8f'l oui ).0i'0­
I. ' H 6.vcU.UTtI(T! A.ucno:; 1 ~0:; j.l1'\ ypaj.lj.lll('i'\o:; l-IepLI(f'lo:; ola<po ptll.:f'lo:;
eE,IO'ooae:wo:; .oC Boussinesq hEl eupe9i'j O~' aUO'tTJ.ptte; 110911.
llatlKi')<; l(aL IpUO'tKi)o:; Ctva)..uO'E.We;. '.t-! A.OO'I~ aim'! rtEplypacpEI
'1'6 IpUO'lI(OV Q)Q\VOj.lEVOV tf'\O:; O'l'payyia£WI; tvao:; ciVOtICTOi}
68poq>opol) atpci:liJatOe; Kelj.l€:vou tnl 6P~OvrlOU liOlanepocHou
UrtoO'fP<l>j.latoo:;, UrtO O'l'paYY10'1'lICGlv 1()..e:lO''1'G:lV 6.ywywv KE11-I€­
vwv UVWaEV toO aotQrtEp6..0l) unoo '1'proI-lQ'wo:; lCai do:; tlJrO­
O"tCl(J1V d ~ 0 6.rt' outoO.
X~210681 ) l1,11 5 'O,366)1, ,7 1)15,21)15 ]"' = 836
,
0,0815,2 1 1,71)
,
d
X
i)1t0A.OY10~oi
.I:U ..TT£pOO'.... aTu
b) 'Arto tilv lE,iO'ooO'lV (30) A.ClIlPUvOllev '1'ilv Tp~'1'nV I(O'1'a rtpoo­
tyYl<J'lV l'I~r,v ri)o:; onoO'rQO'Ero<;
c)
,"pla.o"," A=0,6815
X =2 Io ,6815 ) [I,115(0 ,36 6)14,6~.5)15,195)15 ]" '=
0,081 5,195 - , ,695)
= 83,4 1-'.
(0.742)5= 3 ,7 1~. £IvClI TO veOV (crOOUVClj.lOV pa90e;.
,
= 0.7115. 'A.o TO LX. 2
b) 'Ano l'rlV eE,iO'wa~v (30) A.al-l~QVOl-iev 't'1iv l£tUP'l1v 1C0.ci
npoatYYl(J1V TIIlr,V tf'lo:; unoO''1'uCJt(!Je;
~ 0,742
"ApCl
;:~:~
:,
0a9()(;.
4. Mia t <paPlloyr, ti).; rtapoOOTt<; 9E(J)pioC; etc; rtpalClLKov np6­
PA.llI-IQ O'tpOnlaEW<;, I-Ie l>AOUO:; tOue; 6.ltoPClI'f11touo:; lHtO)..OYl­
O"j.louo:;, lCa910"l'tj; l'T!v l-Ie90oov lCamVOll'1'rlV 1CCl! eUXPllO"1'OV 0q>'
O)"wv tooV QO'X0A.OUIlEVWV I-It npof}A.flj.lClTQ Cl'tpaniO'ewv.
ki)te).oc; unoA.oyuJj.l&v
'H 't'll-lr, 1'00 lO'oO\lVQj.lOU pa90ue; d= 3,695 1-1., ~ Eupegetaa do:;
'1'OV Tpl'1'oV IClUC)..OV tmo)..oYI O'l.lfJ)v, XPllO'll-iOnmeltm ~ '1'11-11'\
'roO d eto:; '1'1\v £siO'uxnv (30) ci:x; Kcitw9l:
napapT111'a
· E~Qywy'l Tilt; t~,awal:wt; TOO
h , d + Y= 3,695 + 1,0 ~ 4,695".
h. = d+ y. = 3,695+ J.5= 5.195".
Ii _ iJh
h
A
8EWpr,O'Wj.lEv 1'6 Cl''1'OIXetov 6YKOU ABr.1.EZH0 (B)". 1:X. A)
Ii
I
ha
1
I
P V _ (}{pv)4:r
ox 2
-J
I
I
I
,.
ax
.;Ix
2
fz
P V + a(pv).dx
I
,II 'r
B
+ {)fr
)
I )
I I
I I
I
I
I
I J-,-----
1
Bounln•• q
ax
V
-----
1
.;Ix
·f
l:x. A.
2
1S6
•Anp().IOt; 1969
IJIlKQUC; ax, )l£CfOU uwouc; h Ko.! ftM::nouc; laou 'fi'\; lJov6:oo~.
~56:<pouC; 6J.lOYEVO~. laotp6-nou Kat K£KOptOj.l(:VOIl 51' uoatur;,
TOi} 6n:o(ou '" ~6.at ~ BrHZ KEinu en;t TO\} 6.0tQ7IEpaTOU 6pl­
....£...
(Kh~)=S
~
OX
ax
at
Cov'[iou lmOOTpciJIJO-rOr; Kal f) KO!III'UA.OYPojJj.wC; bwpnvt:lo.
A6 BE dVaL '" tuu9tpa bUp6.vEHl '(00 tm:oye iou OOOtOC; (b­
Ho)'e in (109)..111) KUttl Tilv XPOVIKf}V crlT'Il.IllV L. -EOTW V 1)
IltO'l 'fQXlrt'lt; po1'lc; El; 16 dvtpO\l toO 6YKOU ABr6EZH0
KCt.t fo'tw P '" 0't"a81lpd nUKv6't'1t; 100 UOQfOl;.
6nou K e:!Vo.l f} uopauA1KT\ 6:,,(IDY11l0'tnc; TOI)
tt511Cl'l no.p oxli 0.01'00 (s pecHic yield).
TIapaoq4L£6a. Ott fl por, toO u1toytiou uoatoe; 6.JC~1A.O ue&i 'tOt;
napa50xac; T(1)\I Dupuit Ko.t Forchh4~illwr: 0.) wO'-OI o.{ ),pal.lIJot
ponr; £Ii; !va. O'OOTllJ.1Q pone; ~aptl''[nTOC; Jtpi)c; Tilv ou::U9UVOlV
/llCl t; 6:j3,a801)c; apvrrmc",c; Jt'l)'f1r; (sink) dvat hp l ~l)V'nm !Cnt p)
°H roxuT."
tb 6:~ouc;
Ko.t S fl
'Exu Sr, oui 1'nV xepintCOO'lv OjJOYEVOl}(; Icot p6Jtou
to6:~oue;
Ka.i OIU ICEKOpEO'IlEVnV POT\v to K Elval CTo.9Ep6v,
(d) ylVE1'al:
n tf)O'WO'IC;
~ ....£... (h~)= ~
S
x
OX
iJ l
f' )
leOta ... f)KOC; "["(j)v YPOJ.lJ.l(1)v poi'jc; dVUI £u8thlt; nvu.­
KA.low f"C; tlt\J9tpcu; t'tI~a.vdac; toO Gon'toe;
Kal li.vEl';apt'ltOe; 'toO paSoue; UUfOO. 'E!papIJo.:;ovtCC; 1'I1V 6.PXl1v
1'i'jc; lha1'np~O' ewc; 't~C; Ilal:nc; btl TOO O'1'olxeiou OYKOU ABr..;o. CZ
He fifOI f} dO'epXOIJEV'l jJlll:o. UOo.tOe; jJEfov tl teEPXoj.d:vl1 Il~a
OOato, 1000l)tal Ilt tf}v llE.ta.~OA.Tlv 'tlie; IJ6.l;;T]e; toD ooo.1'Oe; tVToe;
'tol) tv A.by<!) 6YKO\l, ~jJp6.vojJe:v :
).0)'0'; ltpOC; 1'1'1\1
[
fdl
PV _
afpv)
6%
2
ax
1(h _ allax ax)_
2
- [pv + afpv)
ax2 1(h + ~
~) =
OX
ax2
' E":1'EA.OOVW; 't0J;
A.a:jJj36.v OJ,1EV
6.A.Y E~p lK 6.e; JtPO~ELC;
~
t
de; 1'il'.1 fix; 6voo
ax
ax
fa)
at
"
i)h
-Tx"
fhv! = S ot
fbI
'H oEuttpa D • F nClpaoox i) O'UVF.:1taYEtaL Kal 'n )v toxuv 'toO v6·
IJou 'toi) Darcy, f1Tm
Jerusalem,
Uti!. .
3. nrOQlt. R . H .: (l96~). _D iscussion on UC-III!; U of TIll! Oralllllae
ror FJU ng Water Tab let"j P roc. AfIl . Soc. Civ. Eng. .lou rna l
of 1I'rl.... aM Drainage Ol f ~I on, Vo1. 90, N o. IR. I, MRrcb 196 .
plJ . Il g- I 'lt.
4. VUlI1 m, L.D. : ( 19M), .Nf"w Formula ro r OI'It'rm l"lna Df' p tll
and Spaci ng of :)Ulht.urp.oto Ura i,1S In Irrlga"-' ,l Laurb" Nrrl·
r•.
j~6.
R.li:.: ( 196%), .Ground \Vater l\I ovem('n t. u.S. 1Jt'1I 1
nf tilt l n lc rlor, Ihl re:lU o f Hrrlalll a Uon , \ 'V. alfr R l'lIoureca
"i'echulral Publiql1un , i·:ng1nccrlng i\!onwr aph N o ai, 1 96~.
G/OUlT,
fl . .\ ltIOd1/. W. T . .- {I 96{l), .No nlln t ac' Oirr crrntlal F.'luall on o f
l) raln Spa Ci ng•• Jou rnal o f I rr~ , ll nil Dr:!ln ....c Oh'I!o,lolI . ,\In .
.'joe, Cl'\'. Eng. Vol. 9'2, lR 2, Proc . P:IIl~ :r "8J~, Jun e 19tH), 1~1l .
fe)
7, Poia.!mMflOtln -Karh Ul tl, P. 1'0..: 096'2) . •Th eory or Grou nr1 W :.I ­
Ler M (wt"m~IlL', '1'r:lns l :Ht~1 from RUSSIan I.ly R . J . M. tt.., Wh's t,
Prlnoe llJ n U n! \'l! r~Hy Press. 196 2.
8. V. ,~- . D f'JJ I. of Commr!rU, HandhlJok of Ma t h o.: lll :ltllll'll 10'1111('110 0&
wU11 F(.I... n ul a.~ . (JraI1114. alli l i\lalhe.flI ;ltical Tab l"..8, Na l iOIl<l 1
n il , Ca li or St.andal"l lli, J)~lIc d MathclI lallr J;, !:lcr les ,:;. I !Iii" (p.
251::1).
9. V(I)I ScftilJua.o.r dll, J .. (1 003) . • DC8 1kfl of Tlti! lI m ln14ge j!)r I,'a l ..
HUll Water Tall ks', Jou rnal or I rrkr. and lJralo;I~W U i\" I ~l u n,
AIO . Soc. Ch·. ]~n".. Vol. ~!I, No, l R ~. Proc. P:"PCI' J b''l . JUlie,
I U63, PI> I-II.
10.
'AV1'lKa9uT'l'oovn;c; {liv tf;lO'wmv (c) EL; 'tilv
P6.VO j.lEV:
Scleli tlflC' 'I'rall,slat)on s,
1·9.
'E1ftlOil fI nm;:vOTT]C; P Elval 0''ta.9EPO, fI tf;iO'cooU; (a) yiVE'to.l
V =- K ~
OX
Isrm·J Prlll.:!" r;lOl for
2. 1t()lI-N illClI(J. J . .- ( 19 010) . • R eeh reli C' T h~lJ r iqu CIIIlHr I' CeuuicmenL
d,'. nilpr~ ~ d' O:tlllnmll'(o(~s daltll !t- SOI t!t sur Ie d ;-Illt d es fOUl'­
co,., J O'Jr /l;.1 d e Mal hcltllltlqUf:!I pu r"''' el A~pliqu ~e. , (!:I 6 lIer:lel ),
Tome X·I·'lIliC. I, t 9U 4, pp. !l·U.
cull.ural EnglnceMIlI", Vol. :U, 195>1, p.
t~IO'wO'lv
oh
_ h o fp"1 _ pv fih = p
eX
pS
i . A )'al'w, t·". 1 II'· S . N: ~llm f" rov (196 ~)) . •Th eory or Fluid F low
111 UUfidormulJ)c P orOIl.~ .\ I ed l:! •. Traus latt' tI rrom fl ,iHsl lln.
t.~lO'wmv
(b) ;"0.1l­
Util i Schill,O«1'dc, J ., D . Kirk/lam rt H. K. Frt.""'t'1.; (19M)) ,
.PbyaJca l an a Mat.llc.m a!Jcal 1'ht'(l rif'<' of Till! Itod Ultrh Ural ·
o ate and Llit"ir Viotfolnr.M io I>l'I>leu •. R esc;.u'Ch Bulletin '-l6,
I owa AfI'rlc Exp"'rfmeuL SI •. t UfL
Unsteady drainage of agricultural lands
By George Tenldls *
Summa.. y
T he nonlinea r parti al differential equation of parabolic
lypo
~ht
()
=
~~ -oX
~- (h.£!'..)
cJx
IBoussinesq's eq uali on)
f' )
has bee n found to describ e many physica l prob l(' ms such
as , (aJ th R unsteady bne- diOl I ~fl :; ional now of a co mpres­
iii hi e id eal gas through a hOiJ"logeneous pO I'our; media,
(b) the uns teady one- dimensional fre c surface ground
water flow through a homoge neous porous mcdiu; !e)
the slowly varying unsteady flow in a Hele-:::;haw viscous
fluid mo rel, etc.
In Eq. (1), x is th e horizontal coordinate. t is tho Lime,
· AS!lk!iI.nt
l~ n i ln{·l! r .
nCpl. Qr Waler
Sclen ee
t:lli'l'lI!nll), 01 C..morn!a, lJavls. Ca lif. 956 16.
and
Ilfilil ccrlug.
i
is a cons lant nUUtuer de pending on the porous me­
dium and the fluid, and h is the dependent variable. wi ch
for the CtlSC (a) rt:p rl:f;.mbi the pl'l!Ssure or pressure head
of lh ~ sys Len"l and ror cases (b) and (c) rtrprese nls the dep th
of th e (roc surfaco tro m t he horizo ntal x-axis.
With rererence to Fig. 1, which shows th.· special r.aSO
or seepage in an unconfin ed aquirer with a hori1.untal
imperlll cuble layer and the Lile drains lying on a paraJl el
levd at a di stance d tram th e impermeable layer, Eq. (1 }
i!' ::; ubj ect6d to the following auxiliary conditions:
IniLial co nditions :
h _ ho, l _
x = L.
l _ 0
f2.a)
x ;;' 0,
h ~ hofx l.
BoundaLy co ndiLi ons:
h _ d.
x = 0, 12 .b)
°
~_ O
ax
t> °
'
x = L,
t
~
0
(2.c)
'AllpL\lo<; 1969
157
Soll surfa.ce
7
in which
4
.-;:-.;;;--! ----1---
~,-_._.L.._watel·
I
Tile
------
A
1---1-­
1
I
I
I
I
1
I
lJllperme.ablcl
Ia.yer
I
I
I----L
For valu es o f
355 -16 (
~
>0 .3, Hooghoudt's approximate formula
~
~) + 2 (~)'
(33)
=
[~ (In
; - 115
WI
(34)
Application in drainage design
Given data: d, K, S, Yo. y, a
.i..
h.
a) Calculate
~)
and th en oblai n A from F ig. 2.
b) From Eq. (30) calculate X (fi rs t estimate).
Using the metho d of se paration of veariables for nonli·
near problems, the analytic so luti on of Eqs (1) and (2)
has been round to be
d
X and wh e ther
c.) Calcnl ate
xd- is less
than or greater
than 0.3 use Eq . (33) and Eq. (32) Or Eq . (34), respe­
(29 )
Kht
1
given by the equation
is su ggested, or
Fig . 1. Defillilic/lt uf symhols
h. f (
~
~
f
- - + - L --I
h(x,t) -
the equivalent depth, a is the dr ain radius,
and f is a (un ctio n of
table \
1
d is
ctively , to calc ulate th e equivalent d epth
a
+ t.115A" "'SLo.
d.
by th e ellipti cal
d) Use t his
in stead of d in steps (a) and (b) calculate
the secon d estima te of X.
el Use this new X and the actu al d epth d in ste p (cl to
integral (26), K is the hydraulic con ductivi ty, S is the
specific yield by volume, an d A a [un ctio n defined by Eq .
find a new d. Repeat :>teps (d) an d (e) 2 to 3 times
until therc is no appeciabJe change in X.
in which
r( ~ )
is a Cunction defined
(25) in terms or the incomplete bela-fun ction
Solving Eq. (2 9) ror L and since
r( ~ )
[K
(+ '
+).
,or-~~------------------------.
= 1 at x = L,
one ob tai ns
X ~ 2L ~ 2A [1 .115 K (d
+ y)
S (Y.
(d
+ Y.)
tJ'I.
D.6F-----____-1-..
(30)
y)
41
Eq. (30) may be used to calculate the sp ac ing X or the
tile drains in the case when the deptlI or t he water tab le
Yo from th e tile drains has bee n lowered to dep th y in
time t. The depths Yo and ya re measured at half-distance
be tween t.he drains. Th e numerical value of A ca n be
taken fr om Fig. 2 or rrom equation
(3 1)
aB
which introd uces a n error of less than 3%.
Si nce the diHerential equa tion (-I ) is not valid near the
drains because 01 the convergence of th e s treamlines
(two-dimensional flow ) J an van Schi Hgaarde (1963)
suggested th e use of S. E. Hoogho udt 's ,equivale nt depthll
to repl ace the actual depth. \V. T. Moody {19661 derived
a simplified app roximate formula
",.do = [ 1 ·\
d (8
d
]-1 ,0 ';;--d ';; 0.3
-X
-1't In a. - f
J,
0.'
o'D~~-~----~~--~~--~----~
11
ql
q2
q9
4.1
~6
0,6
",7'
t?8
"'9
d
Fig. i. V(l,[ ues of A and hfJ from Table !.
(32)
TABLE
d
h,'
A
.00
.02
1000
.999
.999
.998
.996
.994
.0/1
.06
.08
.10
-
.1 2
.14
.16
.18
-.991
.98 8
.985
.981
d
d
d
It;
A
.~
A
h,'
A
,20
.22
.24
.26
.28
.30
.32
.34
.36
.38
.977
.972
.967
.962
.956
.948
.40
.42
.44
.46
.48
.50
.908
.898
.888
.S??
.865
.853
.6 0
.62
.64
.66
.68
. 70
.782
.766
.750
.73 2
.7 13
.692
" --­
.6 7'1
.650
.626
.602
-
.942
.933
.926
.917
-
.52
.54
.5 6
.58
-.B40
.8 26
.B13
.797
-
.72
.74
.76
.7 8
d
h,'
.80
.82
.84
.86
.88
.90
A
.575
.5~8
.519
.t187
.453
.414
-.92
- .372
--
.94
.96
.98
1.00
.321
.260
.18 4
0.000
- -­
- -
1,0